Thermal Non-leakage Probability

In an infinite multiplication system, the neutron leakage of the system is logically neglected. But all multiplying systems, all realistic reactor cores are finite multiplying system and this process may not be neglected. During the 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, which is 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.


Pt ~ 0.96


thermal non-leakage probability
The thermal non-leakage probability is for large reactor cores about 0.95 – 0.98 and this value is minimally affected (in comparison with the other factors) by operational changes except 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 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 also depends of the macroscopic slowing down power of neutron moderator (leakage is higher for poor moderators). The diffusion length is given by following equation:

diffusion length

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 following equation:fast non-leakage probability_3For large reactors, we can rewrite this equation without a substantial loss of accuracy simply by replacing the diffusion length Ld and τ by 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 non-leakage probability_4where M is the migration area (m2). The migration length is defined as the square root of the migration area.

Main operational changes, that affect this factor:

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 is the same, the resulting total non-leakage probability is also sensitive on the change in the moderator temperature. In result, an increase in the moderator temperature causes that the probability of leakage increases. 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 and this together causes the increase in the thermal neutron leakage. This physical process is a part of the moderator temperature coefficient (MTC).

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 also on the reload strategy. The neutron leakage is one of key parameters in the neutron and fuel economy.
In order to enhance the neutron and fuel economy, core designers designs 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 highest fuel burnup. These “high” burnup assemblies have inherently lower relative power (due to the lower kinf and due to the fact they feel the presence of non-multiplying environment) in comparison with the average assemblies. In short, this parameter is significantly dependent on the 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 Moderators - Parameters
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.

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