Neutron Leakage

In general, neutrons may leak out of the system during the slowing down process or the neutron diffusion. Neutron leakage can be reduced using reflectors.

In an infinite multiplication system, the leakage of the system is logically neglected. But all multiplying systems, all realistic reactor cores are finite multiplying systems, and the leakage may not be neglected. In general, neutrons may leak out of the system during:

  • During the slowing down process, some of the neutrons leak out of the boundaries of the reactor core before they become thermalized. This process and its impact on the effective multiplying factor is characterized by the fast non-leakage factor, Pf, which is defined as the ratio of the number of fast neutrons that do not leak from the reactor core during the slowing down process to the number of fast neutrons produced by fissions at all energies.
  • two-group-method-reflected-reactor
    This figure shows the general effect of reflection in the thermal reactor system. Note that a reflector can raise the power density of the core-periphery and thus increase the core average power level without changing the peak power.

    During neutron diffusion, some of the neutrons leak out of the boundaries of the reactor core before they are absorbed. This process and its impact on the effective multiplying factor is characterized by the thermal non-leakage factor, Pt, defined as the ratio of the number of thermal neutrons that do not leak from the reactor core during the neutron diffusion process to the number of neutrons that reach thermal energies.

Fast Non-leakage Probability
fast non-leakage probability
Note that there is consistency between the numerator in the definition of ε and the denominator in the definition of Pf.

The fast non-leakage probability is for large reactor cores about 0.92 – 0.98. This value is minimally affected (in comparison with the other factors) by operational changes except for changes in the moderator temperature. It can be derived from the Fermi age theory, the probability that a neutron will remain in the core and become a thermal neutron without being lost by fast leakage is also represented by the following equation:

fast non-leakage probability_2

where τ is fermi age of a neutron, B is the geometrical buckling (in case of critical state Bg = Bm), which depends only on the shape and size of the core. The value of B for small cores is higher than the value for large cores. So that, it is obvious, the fast neutrons leakage is higher for small cores and depends on the macroscopic slowing down the power of neutron moderator (leakage is higher for poor moderators).

Thermal Non-leakage Probability
thermal non-leakage probability
The thermal non-leakage probability is for large reactor cores about 0.95 – 0.98. This value is minimally affected (in comparison with the other factors) by operational changes except for changes in the moderator temperature. The only parameter that influences the thermal non-leakage probability is the moderator temperature. It can be derived from the neutron diffusion theory. The probability that a thermal neutron will remain in the core is also represented by the following equation:

thermal non-leakage probability

in which Ld is the diffusion length, B is the geometrical buckling (in case of critical state Bg = Bm), which depends only on the shape and size of the core. The value of B for small cores is higher than the value for large cores. So that, it is obvious, the thermal neutrons leakage is higher for small cores and depends on the macroscopic slowing down the power of neutron moderator (leakage is higher for poor moderators). The diffusion length is given by the following equation:

diffusion length

Neutron Life Cycle - Thermal Reactor
Six Factor Formula - Four Factor Formula

Total Non-leakage Probability

The fast non-leakage probability (Pf) and the thermal non-leakage probability (Pt) may be combined into one term that gives the fraction of all neutrons that do not leak out of the reactor core. This term is called the total non-leakage probability and is given the symbol PNL, and may be expressed by the following equation:

fast non-leakage probability_3

We can rewrite this equation without a substantial loss of accuracy for large reactors simply by replacing the diffusion length Ld and the fermi age τ with the migration length M in the one group equation. The term B4 is very small for large reactors, and therefore it can be neglected. We may then write.

fast non-leakage probability_4

where M is the migration area (m2), the migration length is defined as the square root of the migration area. As can be seen, the total non-leakage probability of large reactors is primarily a function of the migration area.

Neutron Moderators - Parameters

“Main
Since both (Pf and Pt) are affected by a change in moderator temperature in a heterogeneous water-moderated reactor, and the directions of the feedbacks are the same, the resulting total non-leakage probability is also sensitive on the change in the moderator temperature. As a result, an increase in the moderator temperature causes that the probability of leakage to increase. This effect is one of two main effects causing the moderator temperature coefficient (MTC) of most PWRs to be negative.

The thermal neutron leakage is dependent on the core temperature (or moderator temperature). The moderator temperature influences macroscopic cross-sections for elastic scattering reaction, especially the atomic number density – NH2O(Σss.NH2O) due to the thermal expansion of water. Also, the microscopic cross-section (σa) for neutron absorption changes with core temperature. Both processes have the same direction. As the temperature of the core increases, the diffusion coefficient (D = 1/3.Σtr) increases, and the absorption cross-section decreases, which causes the increase in the thermal neutron leakage. This physical process is a part of the moderator temperature coefficient (MTC).

The fast neutron leakage is also dependent on the core temperature (or moderator temperature). The moderator temperature influences macroscopic cross-sections for elastic scattering reaction (Σss.NH2O) due to the thermal expansion of water. As the temperature of the core increases, the fast neutron leakage increases. This physical process is a part of the moderator temperature coefficient (MTC). It is responsible for an increase in neutron flux measured by neutron detectors situated around the reactor vessel.

In power reactors, the total non-leakage probability also significantly changes with fuel burnup. This dependency is not associated with any of the parameters like the diffusion coefficient or the geometrical buckling. In power reactors, the total non-leakage probability strongly depends on the certain fuel loading pattern, and the reload strategy. Neutron leakage is one of the key parameters in the neutron and fuel economy.

To enhance the neutron and fuel economy, core designers design the low leakage loading patterns, in which fresh fuel assemblies are not situated in the peripheral positions of the reactor core. The peripheral positions are loaded with the fuel with the highest fuel burnup. Compared to the average assemblies, these “high” burnup assemblies have inherently lower relative power (due to the lower kinf and the fact they feel the presence of a non-multiplying environment). In short, this parameter is significantly dependent on a certain loading pattern. During fuel burnup, the neutron leakage usually increases, especially in low leakage loading patterns. This process is caused by reducing the differences in kinf between fresh fuel assemblies and peripheral high-burnup assemblies.

Neutron Reflectors – Leakage Reduction

heavy reflector
Visualization of a heavy reflector. It is only an illustrative example.

It is well known that each reactor core is surrounded by a neutron reflector or reactor core baffle. 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.

Except for research reactors, practically all power reactor cores are designed to minimize neutron leakage. Neutron reflectors surround reactor cores to minimize leakage. The neutron reflector scatters back (or reflects) into the core many neutrons that would otherwise escape. By reducing neutron leakage, the reflector increases keff and reduces the amount of fuel necessary to maintain the reactor critical for a long period. In LWRs, the neutron reflector is installed for the following purposes:

  • The neutron flux distribution is “flattened”, i.e., the ratio of the average flux to the maximum flux is increased. Therefore reflectors reduce the non-uniformity of the power distribution.
  • Because of the higher flux at the edge of the core, there is much better utilization in the peripheral fuel assemblies. In the outer regions of the core, this fuel now contributes much more to the total power production.
  • The neutron reflector scatters back (or reflects) into the core many neutrons that would otherwise escape. The neutrons reflected into the core are available for a chain reaction. This means that the minimum critical size of the reactor is reduced. Alternatively, if the core size is maintained, the reflector makes additional reactivity available for higher fuel burnup. The decrease in the critical size of the core required is known as reflector savings.
  • Neutron reflectors reduce neutron leakage, i.e., to reduce the neutron fluence on a reactor pressure vessel.
  • Neutron reflectors reduce a coolant flow bypass of a core.
  • Neutron reflectors serve as a thermal and radiation shield of a reactor core.
 
References:
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