Effective Multiplication Factor

In this section, the effective multiplication factor, which describes all the possible events in the life of a neutron and effectively describes the state of a finite multiplying system, will be defined.

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 the 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 effective 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 expressed mathematically as shown below.

effective multiplication factor

It is obvious the effective multiplication factor in a multiplying system is a measure of the change in the fission neutron population from one neutron generation to the subsequent generation.

  • keff < 1. If the multiplication factor for a multiplying system is less than 1.0, then the number of neutrons is decreasing in time (with the mean generation time) and the chain reaction will never be self-sustaining. This condition is known as the subcritical state.
  • keff = 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.
  • keff > 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.
The simplest equation governing the neutron kinetics of the system with delayed neutrons is the point kinetics equation. This equation states that the time change of the neutron population is equal to the excess of neutron production (by fission) minus neutron loss by absorption in one mean generation time with delayed neutrons (ld). The role of ld and keff is evident. Larger keff give larger response and longer lifetimes give simply slower responses of multiplying systems.

If there are neutrons in the system at t=0, that is, if n(0) > 0, the solution of this equation gives the simplest point kinetics equation with delayed neutrons (similarly to the case without delayed neutrons):point kinetics equation with delayed neutrons

Let us consider that the mean generation time with delayed neutrons is ~0.085 and k (keff – neutron multiplication factor) will be step increased by only 0.01% (i.e. 10pcm or ~1.5 cents), that is k=1.0000 will increase to k=1.0001.

It must be noted such reactivity insertion (10pcm) is very small in case of LWRs. The reactivity insertions of the order of one pcm are for LWRs practically unrealizable. In this case the reactor period will be:

T = ld / (k-1) = 0.085 / (1.0001-1) = 850s

This is a very long period. In ~14 minutes the neutron flux (and the reactor power) in the reactor would increase by a factor of e = 2.718. This is completely different dimension of the response on reactivity insertion in comparison with the case without presence of delayed neutrons, where the reactor period was 1 second.

Reactors with such a kinetics would be quite easy to control. From this point of view it may seem that reactor control will be a quite boring affair. It will not! The presence of delayed neutrons entails many many specific phenomena, that will be described in later chapters.

Press the “clear and run” button and try to increase the power of the reactor.

Compare the response of the reactor with the case of Infinite Multiplying System Without Source and without Delayed Neutrons (or set the β = 0).

Six Factor Formula

But the effective multiplication factor can be defined also in terms of the most important neutron-physical processes that occur in the nuclear reactor.

There are six factors that describe the inherent multiplication ability of the system. Four of them are completely independent of the size and shape of the reactor and these are:

The fast fission process is in the multiplication factor characterized by the fast fission factor, ε, which increases the fast neutron population in one neutron generation. The fast fission factor is defined as the ratio of the fast neutrons produced by fissions at all energies to the number of fast neutrons produced in thermal fission.
The resonance escape probability, symbolized by p, is the probability that a neutron will be slowed to thermal energy and will escape resonance capture. This probability is defined as the ratio of the number of neutrons that reach thermal energies to the number of fast neutrons that start to slow down.
The thermal utilisation factor, f, is the fraction of the thermal neutrons that are absorbed in the nuclear fuel, in all isotopes of the nuclear fuel. Itdescribes how effectively (how well are utilized) are thermal neutrons absorbed in the fuel.The value of the thermal utilization factor is given by the ratio of the number of thermal neutrons absorbed in the fuel (all nuclides) to the number of thermal neutrons absorbed in all the material that makes up the core.
The number of neutrons created in the new generation is determined by the neutron reproduction factor. The reproduction factor, η, is defined as the ratio of the number of fast neutrons produced by thermal fission to the numberof thermal neutrons absorbed in the fuel.
These factors constitute the infinite multiplication factor (k), which may be expressed mathematically in terms of these factors by following equation, usually known as the four factor formula:

k = η.ε.p.f

The effective multiplication factor (keff) may be expressed mathematically in terms of the infinite multiplication factor (k) and two additional factors which account for neutron leakage during neutron thermalisation (fast non-leakage probability) and neutron leakage during neutron diffusion (thermal non-leakage probability) by following equation, usually known as the six factor formula:

keff = k . Pf . Pt

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.
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.
In reactor physics, keff is the most significant parameter with regard to reactor control. At any specific power level or condition of the reactor, keff is kept as near to the value of 1.0 as possible. At this point in operation, the neutron balance is kept to exactly one neutron completing the life cycle for each original neutron absorbed in the fuel.
The method of calculations of multiplication factors has been developed in the early years of nuclear energy and is only applicable to thermal reactors, where the bulk of fission reactions occurs at thermal energies. This method well puts into the context all the processes, that are associated with the thermal reactors (e.g. the neutron thermalisation, the neutron diffusion or the fast fission), because the most important neutron-physical processes occur in energy regions that can be clearly separated from each other. In short, the calculation of multiplication factor gives a good insight in the processes that occur in each thermal multiplying system.

For fast reactors, in which the fission are caused by neutrons with a very broad energy distribution, such an analysis is inappropriate. The neutron flux in fast reactors have to be divided into many energy groups. Moreover, in fast reactors, the neutron thermalisation is undesirable process and therefore the four factor formula does not really make any sense. The resonance escape probability is not significant because very few neutrons exist at energies where resonance absorption is significant. The thermal non-leakage probability does not exist because the reactor is designed to avoid the thermalization of neutrons.

thermal vs. fast reactor neutron spectrumThe spectrum of neutron energies produced by fission vary significantly with certain reactor design. thermal vs. fast reactor neutron spectrum
Nuclear Fission Chain Reaction
Neutron Life Cycle with keff = 1

Operational factors that affect the multiplication in PWRs.

Detailed knowledge of all possible operational factors that may affect the multiplication factor of the system are of importance in the reactor control. It was stated the keff is during reactor operation kept as near to the value of 1.0 as possible. The criticality of the reactor is influenced by many factors. For illustration, in an extreme case also the presence of human (due to the water, carbon, which are good neutron moderators) near fresh uranium fuel assembly influences the multiplication properties of the assembly.

If any operational factor changes one of the contributing factors to keff (keff = η.ε.p.f.Pf.Pt), the ratio of 1.0 is not maintained and this change in keff makes the reactor either subcritical or supercritical. Some examples of these operational changes, that may take place in PWRs, are below and are described in a separate article in detail.

  • change in the control rods position
  • change in the boron concentration
  • change in the moderator temperature
  • change in the fuel temperature
  • change in the pressure
  • change in the flow rate
  • presence of boiling of the coolant
  • presence of burnable absorbers
  • fuel burnup

See also: Operational changes that affect the multiplication in PWRs.

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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