In typical modern pressurized water reactors (PWRs), the Reactor Coolant System (RCS), shown in the figure, consists of:

• the reactor vessel, which contains the nuclear fuel
• four parallel heat transfer loops connected to a reactor vessel.
• each loop contains a main coolant pump and a steam generator.
• the system includes a pressurizer and its auxiliary systems

All RCS components are located inside the containment building.

At normal operation, there is a compressed liquid water inside the reactor vessel, loops and steam generators.  The pressure is maintained at approximately 16MPa. At this pressure water boils at approximately 350°C (662°F).  Inlet temperature of the water is about 290°C (554°F). The water (coolant) is heated in the reactor core to approximately 325°C (617°F) as the water flows through the core. As it can be seen, the reactor contains approximately 25°C subcooled coolant (distance from the saturation).

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Volumes of typical PWR are in the following table.

It is an illustrative example, following data do not correspond to any reactor design.

This high pressure is maintained by the pressurizer, a separate vessel that is connected to the primary circuit (hot leg) and partially filled with water (partially with saturated steam) which is heated to the saturation temperature (boiling point) for the desired pressure by submerged electrical heaters. Temperature in the pressurizer can be maintained at 350 °C. At normal conditions, about 60% of volume of pressurizer occupies the compressed water and about 40% of volume occupies the saturated steam.

It must be noted the volume of coolant significantly changes with the temperature of the coolant. The total mass of the coolant remains always the same, a change in water volume is not a change in water inventory. The reactor coolant volume changes with temperature because of changes in density. Most substances expand when heated and contract when cooled. However, the amount of expansion or contraction varies, depending on the material. This phenomenon is known as thermal expansion. The change in volume of a material which undergoes a temperature change is given by following relation:

where ∆T is the change in temperature, V is the original volume, ∆V is the change in volume, and α_V is the coefficient of volume expansion.

The volumetric thermal expansion coefficient for water is not constant over the temperature range and increases with the temperature (especially at 300°C), therefore the change in density is not linear with temperature (as indicated in the figure).

See also: Steam Tables

At normal conditions the total volume of coolant in the reactor coolant system is almost constant. On the other hand, during transient load conditions the volume can significantly change. These changes are naturally reflected in a change in pressurizer water level. When the average temperature of reactor coolant goes gradually down, the total water volume is also decreasing, which lowers the pressurizer level. On a gradual load pick-up, the increase in reactor coolant average temperature causes the total water volume to expand, which raises the pressurizer level. These effects must be controlled by pressurizer level control system.

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.

During any power increase the temperature, pressure, or void fraction change and the reactivity of the core changes accordingly. It is difficult to change any operating parameter and not affect every other property of the core. Since it is difficult to separate all these effects (moderator, fuel, void etc.) the power coefficient is defined. The power coefficient combines the Doppler, moderator temperature, and void coefficients. It is expressed as a change in reactivity per change in percent power, Δρ/Δ% power. The value of the power coefficient is always negative in core life but is more negative at the end of the cycle primarily due to the decrease in the moderator temperature coefficient.

Let assume that the reactor is critical at 75% of rated power and that the plant operator wants to increase power to 100% of rated power. The reactor operator must first bring the reactor supercritical by insertion of a positive reactivity (e.g. by control rod withdrawal or borondilution). As the thermal power increases, moderator temperature and fuel temperature increase, causing a negative reactivity effect (from the power coefficient) and the reactor returns to the critical condition. In order to keep the power to be increasing, positive reactivity must be continuously inserted (via control rods or chemical shim). After each reactivity insertion, the reactor power stabilize itself proportionately to the reactivity inserted. The total amount of feedback reactivity that must be offset by control rod withdrawal or boron dilution during the power increase (from ~1% – 100%) is known as the power defect.

Let assume:

• the power coefficient:                 Δρ/Δ% = -20pcm/% of rated power
• differential worth of control rods:    Δρ/Δstep = 10pcm/step
• worth of boric acid:                                      -11pcm/ppm
• desired trend of power decrease:              1% per minute

75% → ↑ 20 steps or ↓ 18 ppm of boric acid within 10 minutes → 85% → next ↑ 20 steps or ↓ 18 ppm within 10 minutes → 95% → final ↑ 10 steps or ↓ 9 ppm within 5 minutes → 100%

Heterogeneity of fuel-moderator assembly. The geometry of the core strongly influences the spatial and energy self-shielding, that take place primarily in heterogeneous reactor cores. 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 (lower) than in the moderator cell due to the high absorption cross-sections of fuel nuclei. This phenomenon causes a significant increase in the resonance escape probability (“p” from four factor formula) in comparison with homogeneous cores.

Reactivity Feedbacks. At power operation (i.e. above 1% of rated power) the reactivity feedbacks causes the flattening of the flux distribution, because the feedbacks acts stronger on positions, where the flux is higher. The neutron flux distribution in commercial power reactors is dependent on many other factors as the fuel loading pattern, control rods position and it may also oscillate within short periods (e.g. as a result of spatial distribution of xenon nuclei). Simply, there is no cosine and J₀ in the commercial power reactor at power operation.

Fuel Loading Pattern. The key feature of PWRs fuel cycles is that there are many fuel assemblies in the core and these assemblies have different multiplying properties, because they may have different enrichment and different burnup. Generally, a common fuel assembly contain energy for approximately 4 years of operation at full power. Once loaded, fuel stays in the core for 4 years depending on the design of the operating cycle. During these 4 years the reactor core have to be refueled. During refueling, every 12 to 18 months, some of the fuel – usually one third or one quarter of the core – is removed to spent fuel pool, while the remainder is rearranged to a location in the core better suited to its remaining level of enrichment. The removed fuel (one third or one quarter of the core, i.e. 40 assemblies) has to be replaced by a fresh fuel assemblies.

A number of different loading patterns have been considered, with the general conclusion that more energy is extracted from the fuel when the power distribution in the core is as flat as possible. In principle, these loading strategies may be divided into two categories:

• Out-In Loading Patterns. In the out-in loading pattern,the fresh fuel batch is placed at the periphery the core, while the intermediate and high burnup batches are placed at the center of the core. At refueling, the highest burnup batch is discharged, the other batches are shifted inward, and a fresh batch is loaded at the periphery. The out-in loading pattern has been found to go too far in the sense that the power distribution is depressed in the center and peaked at the periphery. An additional difficulty is the production of a large number of fast neutrons at the periphery that leak from the core and damage the pressure vessel.
• In-Out Loading Patterns. 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 k_inf and due to the fact they feel the presence of non-multiplying environment – reflector) in comparison with the average assemblies. During fuel depletion, the flux distribution at the periphery of the core increases, especially in low leakage loading patterns. This process is caused by reducing the differences in k_inf between fresh fuel assemblies and peripheral high-burnup assemblies. Since the peripheral assemblies have low relative power, thess loading patterns reach slightly higher peaking factors than Out-In loading patterns.  On the other hand enhanced neutron and fuel economy allows to load less fresh fuel or less enriched fuel during refueling. A secondary benefit is that loading of the “high” burnup assemblies in the periphery reduces the neutron flux on the pressure vessel.  This provides additional protection of the reactor vessel from irradiation embrittlement, caused especially by fast neutrons.

Burnable Absorbers (Burnable Poisons). Burnable absorbers significantly influence the pin-by-pin power distribution. Burnable absorbers are materials that have a high neutron absorption cross-section that are converted into materials of relatively low absorption cross section as the result of radiative capture. Due to the burnup of the absorption material, the negative reactivity of the burnable absorber decreases over core life. Ideally, these absorbers should decrease their negative reactivity at the same rate the fuel’s excess positive reactivity is depleted. In PWRs burnable absorbers are used to decrease initial concentration of boric acid (also to decrease BOC MTC) and to decrease relative power of fresh fuel assemblies. Fixed burnable absorbers are generally used in the form of compounds of boron or gadolinium that are shaped into separate lattice pins or plates, or introduced as additives to the fuel. Since they can usually be distributed more uniformly than control rods, these poisons are less disruptive to the core power distribution.

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  Boron 10. Comparison of total cross-section and cross-section for (n,alpha) reactions.
  Source: JANIS (Java-based Nuclear Data Information Software); The JEFF-3.1.1 Nuclear Data Library

  Boron as Burnable Absorber. In nuclear industry boron is commonly used as a neutron absorber due to the high neutron cross-section of isotope  ¹⁰B. Its (n,alpha) reaction cross-section for thermal neutrons is about 3840 barns (for 0.025 eV neutron). Isotope  ¹¹B has absorption cross-section for thermal neutrons about 0.005 barns (for 0.025 eV neutron). Most of (n,alpha) reactions of thermal neutrons are 10B(n,alpha)7Li reactions accompanied by 0.48 MeV gamma emission. Moreover, isotope ¹⁰B has high (n,alpha) reaction cross-section along the entire neutron energy spectrum. The cross-sections of most other elements becomes very small at high energies as in the case of cadmium. The cross-section of ¹⁰B decreases monotonically with energy. For fast neutrons its cross-section is on the order of barns. Boron as the neutron absorber has another positive property. The reaction products (after a neutron absorption), helium and lithium, are stable isotopes. Therefore there are minimal problems with decay heating of control rods or burnable absorbers used in the reactor core. On the other hand production of helium may lead to significant increase in pressure (under rod cladding), when used as the absorbing material in control rods. Moreover ¹⁰B is the principal source of radioactive tritium in primary circuit of all PWRs (which use boric acid as a chemical shim), because reactions with neutrons can rarely lead to formation of radioactive tritium via:

  • 10B(n,2x alpha)3H                             threshold reaction (~1.2 MeV)
  • 10B(n,alpha)7Li(n,n+alpha)3H     threshold reaction (~3 MeV).

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  Gadolinium 155 and 157. Comparison of radiative capture cross-sections.
  Source: JANIS (Java-based Nuclear Data Information Software); The JEFF-3.1.1 Nuclear Data Library

  Gadolinium as Burnable Absorber. In nuclear industry gadolinium is commonly used as a neutron absorber due to very high neutron absorption cross-section of two isotopes ¹⁵⁵Gd and ¹⁵⁷Gd. In fact their absorption cross-sections are the highest among all stable isotopes. ¹⁵⁵Gd has 61 000 barns for thermal neutrons (for 0.025 eV neutron) and ¹⁵⁷Gd has even 254 000 barns. For this reason gadolinium is widely used as a burnable absorber, which is commonly used in fresh fuel to compensate an excess of reactivity of reactor core. In comparison with another burnable absorbers gadolinium behaves like a completely black material. Therefore gadolinium is very effective in compensation of the excess of reactivity, but on the other hand an improper distribution of Gd-burnable absorbers may lead to unevenness of neutron-flux density in the reactor core.

Presence of Neutron Reflector. The neutron reflector scatters back (or reflects) into the core many neutrons that would otherwise escape (i.e. reduces the neutron leakage). By reducing neutron leakage, the reflector increases k_eff and reduces the amount of fuel necessary to maintain the reactor critical for a long period. 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.

Effect of Fuel Depletion

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_inf will 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 Control Rods

Control rods are an important safety and control system of nuclear reactors. Their prompt action and prompt response of the reactor is indispencable. Control rods are used for maintaining the desired state of chain reaction within a nuclear reactor (i.e. subcritical state, critical state, supercritical state). They constitute a key component of an emergency shutdown system (SCRAM).

At startup mode and at power operation mode control rods are removed from or inserted into the reactor core in order to increase or decrease the reactivity of the reactor (increase or decrease the neutron flux). By the changes of the reactivity the changes of neutron power are performed. This in turn affects the thermal power of the reactor, the amount of steam produced, and hence the electricity generated.

This movement influences the neutron flux distribution radially and axially. The flux depression is naturally higher locally near inserted control rod, but control rods movements also acts globally (e.g. influence axial flux difference).

Effect of Flow Rate

The following effects are valid for pressurized water reactors. Effects of changes in flow rate in boiling reactors are connected with changes in intensity of boiling in channels, which causes these effects are more complex issue.

In PWRs, the effect of change in the flow rate through the primary circuit have significant consequences on the axial power distribution, but in case of PWRs it is not common to change flow rate through the core at power operation.

In reality, when there is an abrupt change (e.g. as a result of a disconnection of the reactor coolant pump) in the flow rate and the reactor power remains the same (e.g. at reduced power), the difference between inlet and outlet temperatures must increase. It follows from basic energy equation of reactor coolant, which is below:

P=↓ṁ.c.↑∆t

The inlet temperature is determined by the pressure in the steam generators, therefore the inlet temperature changes minimally during the transient. It follows the outlet temperature must change significantly as the flow rate changes. When the inlet temperature remains almost the same and the outlet changes significantly, it stands to reason, the average temperature of coolant (moderator) will change also significantly. It follows the temperature of top half of the core increases (in case of flow rate reduction) more than the temperature of bottom half of the core. Since the moderator temperature feedback must be negative, the power from top half will shift to bottom half. Hence the axial flux difference, defined as the difference in normalized flux signals (AFD) between the top and bottom halves of a two section excore neutron detector, will decrease.

The decrease in flow rate is associated with negative reactivity insertion. Special attention is needed in case of an abrupt increase in the flow rate (positive reactivity insertion). At normal operation such increase in the flow rate can not occur, except the controlled reactor coolant pump connection, which can be connected only under specific conditions.

Effect of xenon oscillations

Large thermal reactors with little flux coupling between regions may experience spatial power oscillations because of the non-uniform presence of xenon-135. Xenon-135 is a product of U-235 fission and has a very large neutron capture cross section (about 2.6 x 10⁶ barns). It also decays radioactively with a half-life of 9.1 hours. Little of the Xe-135 results directly from fission, but most comes from the decay chain, Te-135 (β^– decay, 0.5 min) to I-135 (β^– decay, 6.6 hr) to Xe-135. The instantaneous production rate of xenon-135 is dependent on the iodine-135 concentration and therefore on the local neutron flux history. On the other hand, the destruction rate of xenon-135 is dependent on the instantaneous local neutron flux.

The combination of delayed generation and high neutron-capture cross section produces a diversity of impacts on nuclear reactor operation. The mechanism is described in the following four steps.

1. An initial lack of symmetry (let say the axial symmetry in case of axial oscillations) in the core power distribution (for example as a result of significant control rods movement) causes an imbalance in fission rates within the reactor core, and therefore, in the iodine-135 buildup and the xenon-135 absorption.
2. In the high-flux region, xenon-135 burnout allows the flux to increase further, while in the low-flux region, the increase in xenon-135 causes a further reduction in flux. The iodine concentration increases where the flux is high and decreases where the flux is low. This shift in the xenon distribution is such as to increase (decrease) the multiplication properties of the region in which the flux has increased (decreased), thus enhancing the flux tilt.
3. As soon as the iodine-135 levels build up sufficiently, decay to xenon reverses the initial situation. Flux decreases in this area, and the former low-flux region increases in power.
4. Repetition of these patterns can lead to xenon oscillations moving about the core with periods on the order of about 24 hours.

With little change in overall power level, these oscillations can change significantly the local power levels. In a reactor system with strong negative temperature coefficients, the xenon-135 oscillations are damped quite readily. This is one of reasons for designing reactors to have negative moderator-temperature coefficients. Since this effect influences global power distribution in the core, it also influences local power distribution. The problem, however, is in the initial swing of flux levels which displace the flux upward. Since at the higher elevations the local linear heat rate (F_Q(z) limit) is highly restrictive, large xenon spatial oscillations have to be minimized to prevent exceeding F_Q limits.

In order to control xenon spatial oscillations, the axial flux difference or the axial offset are introduced. The most important of these are the axial flux difference (AFD) limits. AFD is a measure of the imbalance between the upper and lower halves of the core in terms of power or flux (ΔI). The AFD is determined  from the outputs of the upper and lower excore neutron detectors, which belong to so called the excore nuclear instrumentation system (NIS).

AFD is defined as:

AFD or ΔI = I_top – I_bottom

where I_top and I_bottom are expressed as a fraction of rated thermal power.

Effect of Thermal Power

The power distribution significantly changes also with changes of thermal power of the reactor. During power changes at power operation mode (i.e. from about 1% up to 100% of rated power) the temperature reactivity effects play very important role. As the neutron population increases, the fuel and the moderator increase its temperature, which results in decrease in reactivity of the reactor (almost all reactors are designed to have the temperature coefficients negative). The negative reactivity coefficient acts against the initial positive reactivity insertion and this positive reactivity is offset by negative reactivity from temperature feedbacks.

This effect naturally occurs on a global scale, and also on a local scale.

During thermal power increase the effectiveness of temperature feedbacks will be greatest where the power is greatest. This process causes the flattening of the flux distribution, because the feedbacks acts stronger on positions, where the flux is higher.

It must be noted, the effect of change in the thermal power have significant consequences on the axial power distribution.

In reality, when there is a change in the thermal power and the coolant flow rate remains the same, the difference between inlet and outlet temperatures must increase. It follows from basic energy equation of reactor coolant, which is below:

P=↓ṁ.c.↑∆t

The inlet temperature is determined by the pressure in the steam generators, therefore the inlet temperature changes minimally during the change of thermal power. It follows the outlet temperature must change significantly as the thermal power changes. When the inlet temperature remains almost the same and the outlet changes significantly, it stands to reason, the average temperature of coolant (moderator) will change also significantly. It follows the temperature of top half of the core increases more than the temperature of bottom half of the core. Since the moderator temperature feedback must be negative, the power from top half will shift to bottom half. In short, the top half of the core is cooled (moderated) by hotter coolant and therefore it is worse moderated. Hence the axial flux difference, defined as the difference in normalized flux signals (AFD) between the top and bottom halves of a two section excore neutron detector, will decrease.

AFD is defined as:

AFD or ΔI = I_top – I_bottom

where I_top and I_bottom are expressed as a fraction of rated thermal power.