Fuel Burnup - Definition and Calculations

Contents

Fuel Burnup – Core Burnup
Units of Fuel Burnup
Fuel Depletion – Isotopic Changes
Nuclear Transmutation
Evolution Equations
Isotopic Changes – Summary
Neutron Poisons – Reactor Poisoning
Reactor Slagging – Reactor Slag
Effect of Fuel Depletion on the Power Distribution
Effect of Fuel Depletion on Delayed Neutrons
Reactivity Effects of Fuel Burnup
Neutron Flux Effects of Fuel Burnup
LWR with High Burnup Fuel

Fuel Burnup – Core Burnup

In nuclear engineering, fuel burnup (also known as fuel utilization) is a measure of how much energy is extracted from a nuclear fuel and a measure of fuel depletion. The most commonly defined as the fission energy release per unit mass of fuel in megawatt-days per metric ton of heavy metal of uranium (MWd/tHM), or similar units. Fuel burnup defines energy release as well as it defines isotopic composition of irradiated fuel. Since during refueling, every 12 to 18 months, some of the fuel – usually one third or one quarter of the core – is replaced by a fresh fuel assemblies and power distribution is not uniform in the core, reactor engineers distinguish between:

Core Burnup. Averaged burnup over entire core (i.e. over all fuel assemblies). For example – BU_core = 25 000 MWd/tHM
Fuel Assembly Burnup. Averaged burnup over single assembly (i.e. over all fuel pins of a single fuel assembly). For example – BU_FA = 40 000 MWd/tHM
Pin Burnup. Averaged burnup over single fuel pin or fuel rod (over all fuel pellets of a single fuel pin). For example – BU_pin = 45 000 MWd/tHM
Local or Fine Mesh Burnup. Burnup significantly varies also within single fuel pellet. For example, the local burnup at the rim of the UO₂ pellet can be 2–3 times higher than the average pellet burnup. This local anomaly causes formation of a structure known as High Burnup Structure.

Units of Fuel Burnup

There are three different units in common use for describing the state or depletion of the fuel. These units are:

Megawatt-day per Metric Ton of Heavy Metal (unit: MWd/tHM)

In general, megawatt-day is a derived unit of energy. It is used to measure energy produced, especially in power engineering. One megawatt-day is equal to one megawatt of power produced by power plant over a period of one day (megawatts multiplied by the time in days). 1 MWd = 24,000 kWh.

At nuclear power plants there are also gigawatt-days, because it approximately corresponds to energy produced by power plant over a period of one day. This unit (MWd) was also used to derive unit of fuel burnup. The most commonly used measure of fuel burnup is the fission energy release per unit mass of fuel. Therefore fuel burnup of nuclear fuel is normally have units of megawatt-days per metric tonne (MWd/MTU), where tonne refers to a metric ton of uranium metal (sometimes MWd/tU HM as Heavy Metal). In this field, the megawatt-day refers to the thermal power of the reactor, not the fraction which is converted to electricity. For example, for a typical nuclear reactor with a thermal power of 3,000 MWth, about ~1,000 MWe of electrical power is generated in the generator.

For example, a reactor with 100,000 kg of fuel operating at 3000 MWth power level for 1,000 days would have a burnup increase of 30,000 MWd/MTU. As was written, each watt of power production requires about 3.1×10¹⁰ fissions per second. In words of fissions, fissioning of about 1 g of U-235 produces about 1 MWd of thermal energy (see: Energy Release per Fission).

Discharged fuel (i.e. after four years of operation) from light water reactors has usually a burnup of 45,000 to 50,000 MWd/tU. This means that about 45 to 50 kg of fissile material per metric ton of nuclear fuel used have been fissioned.

Burnup Fraction - Fission Burnup

Engineers sometimes use fission burnup or the burnup fraction as a useful unit of fuel burnup. This unit expresses the number of fission normalised to the initial number of uranium nuclei (or another fissionable nuclei). Fission burnup is the proportion of heavy nuclei placed in the reactor core that have undergone nuclear fission either directly or after conversion. In general 1% in fission burnup is approximately equal to 10,000 MWd/tU.

Therefore, PWRs, which have an initial load of about 4% of uranium 235 (96% of uranium 238), reach about 4% of burnup fraction.

EFPD - Effective Full Power Day

boron concentration vs. cycle burnup - PWR — boron concentration vs. cycle burnup – PWR

Commonly used unit, that describes the state of the fuel is the Effective Full Power Day (EFPD). It specifies the burnup of a given fuel assembly by the number of days it has resided in the core while the core was operated at full power. This unit is commonly used especially for cycle burnup description. For example, a typical 18 month fuel cycle requires an excess of reactivity for 500 EFPDs. It means the core must be refueled after 500 days at 100% of rated power, or after 1000 days at 50% of rated power. The relation between EFPD and MWd/tU is:

Neutron Fluence

In nuclear engineering, we have to distinguish between the neutron flux density, the neutron intensity and the neutron fluence.

Neutron fluence, previously referred to as the neutron dose, is defined as the time integral of the neutron flux density, expressed as number of particles (neutrons) per cm². Neutron fluence is primarily defined for material engineering, but is widely used by reactor engineers as a unit of fuel burnup.

Neutron Fluence and Fuel Burnup – Neutrons per Kilobarn

Neutron fluence can be used as a measure of fuel burnup as well. Since reaction rate is given by the product RR = Ф . Σ, the rate of burnup is proportional to the neutron flux. The accumulated burnup over a specific period of time (t) is therefore proportional to the product of flux and time (F = Ф . t or F = ∫Фdt). This product is known as the neutron fluence or the total neutron exposure of the fuel. The units of neutron fluence are also neutrons per m², but, in practice, it is often expressed in neutrons per kilobarn:

1 n/kb = 10²⁵ neutrons per m²

For example, let assume the fuel in PWR, which is irradiated with a flux on the order of 3×10¹⁷ neutrons.m^-2s^-1 for approximately 4 years. The total fluence of discharged fuel is then 4 n/kb.

Neutron Fluence and Irradiation Embrittlement

During the operation of a nuclear power plant, the material of the reactor pressure vessel and the material of other reactor internals are exposed to neutron radiation (especially to fast neutrons), which results in localized embrittlement of the steel and welds in the area of the reactor core. Irradiation embrittlement can lead to loss of fracture toughness. Typically, the low alloy reactor pressure vessel steels are ferritic steels that exhibit the classic ductile-to-brittle transition behaviour with decreasing temperature. This transitional temperature is of the highest importance during plant heatup.

Failure modes:

Low toughness region: Main failure mode is the brittle fracture (transgranular cleavage). In brittle fracture, no apparent plastic deformation takes place before fracture. Cracks propagate rapidly.
High toughness region: Main failure mode is the ductile fracture (shear fracture). In ductile fracture, extensive plastic deformation (necking) takes place before fracture. Ductile fracture is better than brittle fracture, because there is slow propagation and an absorption of a large amount energy before fracture.

Neutron irradiation tends to increase the temperature (ductile-to-brittle transition temperature) at which this transition occurs and tends to decrease the ductile toughness.

Since the reactor pressure vessel is considered irreplaceable, neutron irradiation embrittlement of pressure vessel steels is a key issue in the long term assessment of structural integrity for life attainment and extension programmes.

Radiation damage is produced when neutrons of sufficient energy displace atoms (especially in steels at operating temperatures 260 – 300°C) that result in displacement cascades which produce large numbers of defects, both vacancies and interstitials. Although the inside surface of the RPV is exposed to neutrons of varying energies, the higher energy neutrons, those above about 0.5 MeV, produce the bulk of the damage. In order to minimize such material degradation type and structure of the steel must be appropriately selected. Today it is known that the susceptibility of reactor pressure vessel steels is strongly affected (negatively) by the presence of copper, nickel and phosphorus.

To minimize neutron fluence:

Radial neutron reflectors are installed around the reactor core. Neutron reflectors reduce neutron leakage and therefore they reduce the neutron fluence on a reactor pressure vessel.
Core designers design the low leakage loading patterns, in which fresh fuel assemblies are not situated in the peripheral positions of the reactor core.

Heating the irradiated steel to a temperature sufficiently above the irradiation temperature can mitigate the embrittlement. At normal operation of LWRs, the RPV material temperature is far from this temperature. Therefore, when it is required, plant operators must perform thermal annealing of irradiated reactor pressure vessel material to restore the mechanical properties. The degree of recovery for a given steel depends on the time and the temperature at which annealing is performed.

See also: Integrity of reactor pressure vessels in nuclear power plants: assessment of irradiation embrittlement effects in reactor pressure vessel steels. International Atomic Energy Agency, ISBN 978-92-0-101709-3, Vienna, 2009.

Fuel Depletion – Isotopic Changes

As a reactor is operated at significant power, atoms of fuel are constantly consumed, resulting in the slow depletion of the fuel. It must be noted there are also research reactors, which have very low power and the fuel in these reactors does not change its isotopic composition.

Research reactors with significant thermal power and all power reactors are subjected to significant isotopic changes. The study of these isotopic changes is known as the long-term kinetics, which describes phenomena that occur over months or even years. The study of phenomena that occur over the course of several hours to a few days, for example, effects of neutron poisons on the reactivity (i.e. Xenon poisoning or spatial oscillations), is known as the medium-term kinetics.
This chapter describes the long-term kinetics of thermal reactors based on the uranium fuel cycle, in which a fuel with a large concentration of uranium-238 is used (e.g. PWRs or BWRs). Most common reactor fuels are composed of either natural or partially enriched uranium. Typically, PWRs uses an enriched uranium fuel (~4% of U-235) as a fresh fuel. Exposure to neutron flux gradually depletes the uranium-235, decreasing core reactivity (compensated by control rods, chemical shim or burnable absorbers). The initial fuel load of a new reactor core (so called first core) is entirely fresh fuel, that is, fuel with no plutonium or fission products present. The contribution of uranium-238 directly to fission is quite small in most thermal reactors. On the other hand uranium-238 plays very important role and this chapter is primarily about this isotope.

Nuclear Transmutation

In physics, nuclear transmutation is the conversion of one chemical element or an isotope into another. In nuclear engineering, also nuclear reactors cause artificial transmutation by exposing elements to neutrons produced by fission. In nuclear reactors the most important phenomenon is the transmutation of fertile isotopes (e.g. uranium 238) into fissile isotopes (e.g. plutonium 239). The process of the transmutation of fertile materials to fissile materials is referred to as fuel breeding.

Fertile materials are not capable of undergoing fission reaction after absorbing thermal (slow or low energy) neutrons and these materials are not capable of sustaining a nuclear fission chain reaction. There are two basic fertile materials: ²³⁸U and ²³²Th.

²³³U breeding:

$n+_{90}^{232}\textrm{Th} {\rightarrow} _{90}^{232}\textrm{Th}+\gamma \rightarrow _{91}^{233}\textrm{Pa} \rightarrow _{92}^{233}\textrm{U}$

²³²Th is the predominant isotope of natural thorium. If this fertile material is loaded in the nuclear reactor, the nuclei of ²³²Th absorb a neutron and become nuclei of ²³³Th. The half-life of ²³³Th is approximately 21.8 minutes. ²³³Th decays (negative beta decay) to ²³³Pa (protactinium), whose half-life is 26.97 days. ²³³Pa decays (negative beta decay) to ²³³U, that is very good fissile material. On the other hand proposed reactor designs must attempt to physically isolate the protactinium from further neutron capture before beta decay can occur.

²³⁹Pu breeding:

$n+_{92}^{238}\textrm{U} {\rightarrow} _{92}^{239}\textrm{U}+\gamma \rightarrow _{93}^{239}\textrm{Np} \rightarrow _{94}^{239}\textrm{Pu}$

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. ²³⁹U decays (negative beta decay) to ²³⁹Np (neptunium), whose half-life is 2.36 days. ²³⁹Np decays (negative beta decay) to ²³⁹Pu.

Transmutation of transuranium elements (i.e. the chemical elements with atomic numbers greater than 92 ) such as the isotopes of plutonium has the potential to help solve some problems posed by the management of radioactive waste by reducing the proportion of long-lived isotopes it contains.

Evolution Equations

The exact evolution of isotopic changes is usually modelled mathematically a set of differential equations known as evolution equations. These equations describe the rate of burnup of U-235, the rate of buildup of Pu-239, production of Pu-240 and Pu-241, the buildup of neutron absorbing fission products, and the overall rate of reactivity change in the reactor due to the changing composition of the fuel. The evolution equation can be constructed for each isotope. For example:

Special reference: W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.

Special reference: Paul Reuss, Neutron Physics, EDP Sciences, 2008, ISBN: 2759800415.

Isotopic Changes – Summary

In summary, it can be seen for fuel burnup of 40 GWd/tU:

Approximately 3 – 4% of the heavy nuclei are fissioned.
About two thirds of these fissions come directly from uranium 235, and the other third from plutonium, which is produced from uranium 238. The contribution significantly increases as the fuel burnup increases.
The removed fuel (spent nuclear fuel) still contains about 96% of reusable material. It must be removed due to decreasing k_inf of an assembly or in or in other words, it must be removed due to accumulation of fission products with significant absorption cross-section.
Discharged fuel contains abour 0.8% of plutonium and about 1% of uranium 235. It must be noted, there is a significant content (about 0.5%) of uranium 236, which is is neither a fissile isotope, nor a fertile isotope.

Neutron Poisons – Reactor Poisoning

Fission fragment yields — Fission fragment yield for different nuclei. The most probable fragment masses are around mass 95 (Krypton) and 137 (Barium).

Nuclear fission fragments are the fragments left after a nucleus fissions. The average of the fragment mass is about 118, but very few fragments near that average are found. It is much more probable to break up into unequal fragments, and the most probable fragment masses are around mass 95 (Krypton) and 137 (Barium). Most of these fission fragments are highly unstable (radioactive) and undergo further radioactive decays to stabilize itself.

Plenty of isotopes, usually neutron rich isotopes, are produced by fission. Many of the resulting fission products have measurable thermal absorption cross sections. Fission products are of concern in reactors primarily because they become parasitic absorbers of neutrons and result in long term sources of heat (so called decay heat). Their buildup in the reactor tends to reduce the multiplication factor. For that reason, these nuclei are known as neutron poisons.

The significance of each poison depends both on the yield, decay rates and its absorption cross-section. During reactor operation equilibrium is eventually reached if a radioisotope is produced at a constant rate, for after several half-lives the rate of decay will equal the rate of production. Most fission products have low absorption cross-section, but there are very important exceptions. These isotopes are described in the following table.

As can be seen, most of these isotopes are stable and some undergo a decay. From this point of view, we distinguish between neutron poisons, which causes reactor poisoning and neutron poisons, which causes reactor slagging.

The isotope with short half-life (e.g. xenon-135) is known as reactor poison, while the isotope that is long-lived or even stable is known as reactor slag. The neutron absorbing fission products xenon-135 and samarium-149 have particular operational importance.

Their concentrations can change quickly, producing major changes in neutron absorption on a relatively short time scale. Because these two fission product poisons remove neutrons from the reactor, they will have an impact on the thermal utilization factor and thus k_eff and reactivity. Importance of xenon-135 and samarium-149 is so high, that it will be discussed in separate section.

It must be noted, neutron-absorbing materials, also called neutron poisons, are intentionally inserted into some types of reactors in order to lower the high reactivity of their initial fresh fuel load. Some of these poisons deplete as they absorb neutrons during reactor operation, while others remain relatively constant.

Reactor Slagging – Reactor Slag

As was written, an isotope that is long-lived or even stable is known as reactor slag. The accumulation of these parasitic absorbers is known as reactor slagging and determines the lifetime of nuclear fuel in a reactor, since their buildup causes (together with a decrease in fissile material) continuous decrease in core reactivity.

Samarium 149 belongs to this group of isotopes, but its importance is so high, that it is usually discussed separately. As can be seen from the table, there are numerous other fission products that, as a result of their concentration and thermal neutron absorption cross section, have a poisoning effect on reactor operation. Other fission products with relatively high absorption cross sections include ⁸³Kr, ⁹⁵Mo, ¹⁴³Nd, ¹⁴⁷Pm. Individually, they are of little consequence, but taken together they have a significant effect. These are often characterized as lumped fission product poisons.

As was written, for fuel burnup of 40 GWd/tU, approximately 3 – 4% of the heavy nuclei are fissioned. The discharged fuel (spent nuclear fuel) still contains about 96% of reusable material. It must be removed due to decreasing k_inf of an assembly or in or in other words, it must be removed due to:

accumulation of fission products with significant absorption cross-section.
decrease in fissile material

This is the reason, why some nuclear power plant operators use reprocessed fuel to partially replace fresh uranium fuel. Nuclear reprocessing is technology that was developed to chemically separate and recover fissionable material from spent nuclear fuel as well as to remove these isotopes with significant absorption cross-section.

Effect of Fuel Depletion on the Power Distribution

During operation of a reactor the amount of fissile material contained in the fuel assembly constantly decreases, therefore the assembly k_inf constantly decreases. During fuel depletion the decrease of the assembly k_infwill be greatest where the power is greatest. The differences in k_inf between fresh fuel assemblies and high-burnup assemblies decreases. Therefore during cycle depletion, this process will cause the power to shift away from regions with highest k_inf. This process also depends on the use of burnable absorbers, which disrupt the first assumption about the constantly decreasing assembly k_inf.

Effect of Fuel Depletion on Delayed Neutrons

In LWRs the delayed neutron fraction decreases with fuel burnup. This is due to isotopic changes in the fuel. It is simple, fresh uranium fuel contains only ²³⁵U as the fissile material, meanwhile during fuel burnup the importance of fission of ²³⁹Pu increases (in some cases up to 50%). Since ²³⁹Pu produces significantly less delayed neutrons (0.0021 vs. 0.0064), the resultant core delayed neutron fraction of a multiplying system decreases (it is the weighted average of the constituent delayed neutron fractions). It follows then that the amount of reactivity insertion needed to produce a given reactor period decreases with burnup of the fuel. This is also the reason why the neutron spectrum in the core become harder with fuel burnup.

β_core= ∑ P_i.β_i

where P_i is fraction of power generated by isotope i.

Reactivity Effects of Fuel Burnup

↑burnup ⇒ ↓k_eff = ↓η .ε.p. ↓f . ↑↓P_f. ↑↓P_t

It is hard to describe the effects of fuel burnup on the six factor formula. It must be noted, the criticality must be maintained for long period and therefore all the negative effects must be compensated by the increase in the thermal utilization factor (boron dilution or compensating rods withdrawal).

See more: Operational factors that affect the multiplication

Neutron Flux Effects of Fuel Burnup

In a power reactor over a relatively short period of time (days or weeks), the atomic number density of the fuel atoms remains relatively constant. Therefore in this short period, also the average neutron flux remains constant, when reactor is operated at a constant power level. On the other hand, the atomic number densities of fissile isotopes over a period of months decrease due to the fuel burnup and therefore also the macroscopic cross-sections decrease. Note that, the thermal power is determined by reaction rate (RR = Ф . Σ), in which Σ is continuously decreasing. This results is slow increase in the neutron flux in order to keep the desired power level.

LWR with High Burnup Fuel

Current light water reactors are typically designed to achieve burnup about 50 GWd/tU. With newer fuel technology, and particularly the use of advanced burnable absorbers, these same reactors are now capable of achieving up to 60 GWd/tU. Some studies shows that in the near future, even with the present enrichment limit (5 wt %), fuel burnup could be extended near to 70 MWd/kg. Further increase in fuel burnup is impossible without the relaxation of the present enrichment limit (5%).

As can be seen, there has been a clear trend over the last decades toward increasing fuel burnup in light water reactors. In nuclear power plants, high fuel burnup is desirable for:

Reducing the number of fresh nuclear fuel assemblies required and thus reducing spent nuclear fuel assemblies generated. Both aspects lead to improvements in the economics of fuel cycle.
Reducing duration of refueling outage.
High burnup results in a lower mass of spent fuel discharged per unit of electricity generated. This can reduce spent fuel handling and transportation.
A benefit (crutial aspect for some operators) is that loading of the “high” burnup assemblies in the periphery reduces the neutron flux on the pressure vessel.
Reducing the potential for diversion of fissile material from spent fuel for non-peaceful.

On the other hand, there are signals, that increasing burnup above 50 or 60 GWd/tU leads to significant engineering challenges and even it does not necessarily have to lead to economical benefits.

See also: Etienne Parent. Nuclear Fuel Cycles for Mid-Century Deployment. MIT. 2009.

Higher burnup fuels require higher initial enrichment to sustain reactivity. Since the amount of separative work units (SWUs) is not a linear function of enrichment, it is more expensive to enrich higher enrichments. For instance, to produce one kilogram of uranium enriched to:

5% U-235 requires 8.9 SWU if the tails assay is 0.20%
4% U-235 requires 6.3 SWU if the tails assay is 0.20%

But there are also operational aspects of high burnup fuels that are associated especially with reliability of such fuel. Main concerns associated with high burnup fuels are:

Increase burnup places additional demands on fuel cladding, which must withstand in the reactor environment for longer period.
The longer residence in the reactor requires a higher corrosion resistance.
Higher burnup leads to higher accumulation of gaseous fission products inside the fuel pin resulting in significant increase in internal pressure.
Higher burnup leads to increased radiation induced growth, that can lead to undesirable changes in core geometry (fuel assembly bow or fuel rod bow). FA bow can result in an increased control rods drop time due to friction between control rod and bowed guide tubes.
These factors must be considered, because they can lead to a decrease in operational reliability
High burnup fuel generates a smaller volume of fuel for reprocessing, but with a higher specific activity.

Strategic management and decision making in respect of the middle part of nuclear fuel cycles is a very specific problem of power engineering. It is obvious also such evaluation must contain nuclear calculations and business-economic evaluation, thus they cannot be carried out separately. Detailed nuclear calculations are necessary, because they can exclude many promising strategies. On the other hand they do not provide any information about benefits or costs. A strategic decision making must be carried out on the basis of business-economic evaluations.

Special reference: Technical and economic limits to fuel burnup extension, Proceedings of a Technical Committee meeting, IAEA, 11/2009

References:

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J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
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K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
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E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.

See above:

Reactor Operation