Thermal Utilization Factor

It is obvious, the neutrons that escape the resonance absorption and remain in the core will be thermalized. In thermal reactors these neutrons continue to diffuse throughout the reactor, until they are absorbed. But there is many materials in the reactor core, in which these neutrons may be absorbed. The thermal utilization factor, f, is the fraction of the thermal neutrons that are absorbed in the nuclear fuel, in all isotopes of the nuclear fuel. It describes 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.


f ~ 0.70


thermal utilisation factor

From definition, this factor will always be less than one, because there will be always some of the thermal neutrons absorbed in non-fuel materials (usually referred to as a parasitic absorption). For homogenous reactor cores, this factor can be calculated in terms of macroscopic cross-sections by the following equation:

thermal utilisation factor - equation

where Σa is the macroscopic absorption cross section, which is the sum of the capture cross section and the fission cross section, Σa = Σc + Σf. The superscripts U, M, P, CR, B, BA and O, refer to uranium fuel, moderator, poisons, control rods, boric acid, burnable absorbers and others, respectively. It is obvious, that the presence control rods, boric acid or poisons causes a decrease in the neutron utilization, which, in turn, causes a decrease of multiplication factor.

In heterogeneous reactor core the calculation is more complicated. The thermal utilization factor must be expressed in terms of reaction rates as a consequence self-shielding effects. In short, the neutron flux is not constant due to the heterogeneous geometry of the unit cell. The flux will be different in the fuel cell than in the moderator cell due to the high absorption cross-sections of fuel nuclei. Moreover, the atomic number densities and volumes of fuel, moderator and poisons will be different due to the heterogeneous geometry of the unit cell. Therefore the neutron utilization factor for heterogeneous reactor cores must be calculated in terms of reaction rates and volumes, for example, by the following equation:

thermal utilisation factor - equation2

In heterogeneous reactors there is usually some difference between the neutron flux in the moderator and the neutron flux in the fuel. This difference is caused by the high absorption in the fuel and the ratio of thermal neutron flux in the moderator to that in the fuel is known as the thermal disadvantage factor defined as:

disadvantage factor

The thermal utilization factor is for heterogeneous reactor cores about 0.5 – 0.8, but it depends strongly on the reactor type.

Main operational changes, that affect this factor:

This factor is the key factor, that the reactor operator has the greatest control over. Control rods insertion/withdrawal and to borate/boron dilution from the moderator are operations, which influences precisely the thermal utilization factor. For example, control rods insertion causes an addition of new absorbing material into the core and this causes a decrease in thermal utilization factor.

The thermal utilization factor slightly changes with the fuel burnup. The fresh fuel at the beginning of the cycle comprises only the absorption by the 235U. As the amount of 239Pu and other higher transuranic elements increases because of the radiative capture of neutron by the 238U in the core, it is necessary to consider the change of fuel composition in determining the value of f at different times of the fuel cycle.

In general, the thermal utilization factor decreases in time as the total content of fissile isotopes decreases and the total content of neutron poisons (fission products with high absorption cross-sections) increases. But in the power reactors, in which the criticality must be maintained for long period (e.g. 12-month or up to 24-month) without refueling, the thermal utilization factor may not decrease. Such the decrease would imply inevitable reactor shutdown. The continuous decrease in ΣaU must be offset by the continuous decrease in ΣaB, which means the concentration of boric acid (in case of PWRs) must be continuously decreased as the fuel loses its reactivity (kinf). For reactors, in which the chemical shim can not be used, the excess of reactivity is compensated by compensating rods.

On the other hand, there is the fuel breeding, which plays very important role in the compensation of decreasing 235U content.
Fuel breeding chain.
Chain of the transuranic elements.
Boron letdown curve.
Boron letdown curve (chemical shim) and boron 10 depletion during a 12-month fuel cycle.
All commercial light water reactors contains both fissile and fertile materials. For example, most PWRs use low enriched uranium fuel with enrichment of 235U up to 5%. Therefore more than 95% of content of fresh fuel is fertile isotope 238U. During fuel burnup the fertile materials (conversion of 238U to fissile 239Pu known as fuel breeding) partially replace fissile 235U, thus permitting the power reactor to operate longer before the amount of fissile material decreases to the point where reactor criticality is no longer manageable.

The fuel breeding in the fuel cycle of all commercial light water reactors plays a significant role. In recent years, the commercial power industry has been emphasizing high-burnup fuels (up to 60 – 70 GWd/tU), which are typically enriched to higher percentages of 235U (up to 5%). As burnup increases, a higher percentage of the total power produced in a reactor is due to the fuel bred inside the reactor.

At a burnup of 30 GWd/tU (gigawatt-days per metric ton of uranium), about 30% of the total energy released comes from bred plutonium. At 40 GWd/tU, that percentage increases to about forty percent. This corresponds to a breeding ratio for these reactors of about 0.4 to 0.5. That means, about half of the fissile fuel in these reactors is bred there. This effect extends the cycle length for such fuels to sometimes nearly twice what it would be otherwise. MOX fuel has a smaller breeding effect than 235U fuel and is thus more challenging and slightly less economic to use due to a quicker drop off in reactivity through cycle life.

Plutonium 239 breeding

 Equation - Plutonium 239 breeding from Uranium 238 

Neutron capture may also be used to create fissile 239Pu from 238U, which is the dominant constituent of naturally occurring uranium (99.28%). Absorption of a neutron in the 238U nucleus yields 239U. The half-life of 239U is approximately 23.5 minutes. 239decays (negative beta decay) to 239Np (neptunium), whose half-life is 2.36 days. 239Np decays (negative beta decay)  to 239Pu.

moderator-to-fuel ratioThe thermal utilization factor is dependent also on the moderator-to-fuel ratio – NM / NF. With the change of the moderator-to-fuel ratio changes also the absorption by hydrogen atoms in the reactor core and simply changes probability of absorption by fuel atoms.

Much more important, however, is the reactor design, which determines the moderator-to-fuel ratio. Most of light water reactors are designed as so called undermoderated with the slightly harder neutron flux spectrum (the moderation is slightly insufficient) than in an optimum case.

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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