Power Distribution in Conventional Reactor Cores

Solution for finite cylindrical homogenous reactor.

Solution for finite cylindrical homogenous reactor.

It should be noted the flux shape derived from the diffusion theory is only a theoretical case in a uniform homogeneous cylindrical reactor at low power levels (at “zero power criticality”). We have implicitly assumed that the core consisting of thousands of fuel and control elements, coolant, and structure can be represented by some effective homogeneous mixture. This is a very strong assumption, because it does not take into account the heterogeneity of a core.

See also: Diffusion Equation – Finite Cylindrical Reactor

Heterogeneous Core

heterogeneous-core-minMost of PWRs use the uranium fuel, which is in the form of uranium dioxide. Uranium dioxide is a black semiconducting solid with very low thermal conductivity. On the other hand the uranium dioxide has very high melting point and has well known behavior. The UO2 is pressed into pellets, these pellets are then sintered into the solid.

These pellets are then loaded and encapsulated within a fuel rod (or fuel pin), which is made of zirconium alloys due to its very low absorption cross-section (unlike the stainless steel). The surface of the tube, which covers the pellets, is called fuel cladding. The collection of fuel rods or elements is called the fuel assembly. The fuel assembly constitute the base element of the nuclear reactor core. The reactor core (PWR type) contains about 157 fuel assemblies (depending on a reactor type). Western PWRs use a square lattice arrangement and assemblies are characterized by the number of rods they contain, typically, 17×17 in current designs. The enrichment of fuel rods is never uniformed. The enrichment is differentiated in radial direction but also in axial direction. This arrangement significantly influences the power distribution.

Russian VVER-type reactors use a fuel that is characterized by their hexagonal arrangement, but is otherwise of similar length and structure to other PWR fuel assemblies.

Flux Distribution

In commercial reactor cores the flux distribution is significantly influenced by:

flux of resonance neutronsHeterogeneity 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.

Power Distribution - Nuclear Reactor

In commercial reactor cores the flux distribution is significantly influenced by many factors. 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 kinf 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 kinf 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.

  • 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 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(n,alpha) reactions of 10BMoreover, 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).
  • Gadolinium 155 and 157. Comparison of radiative capture cross-sections.

    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.

effect of gadolinium absorbers

The effect of gadolinium burnable absorbers (BA) can be demonstrated on boron letdown curves. At the beginning of specific fuel cycle the critical concentration of boric acid in the reactor core without burnable absorbers (blue curve) significantly differs from the critical concentration of boric acid in the reactor core with burnable absorbers (red curve). The difference is dependent on the amount of BA used.

two-group-method-reflected-reactor

This figure shows the general effect of reflection in the thermal reactor system. Note that a reflector can raise the power density of the core periphery and thus increase the core average power level without changing the peak power.

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 keff 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.

Operational factors that affect 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 kinf constantly decreases. During fuel depletion the decrease of the assembly kinf will be greatest where the power is greatest. The differences in kinf 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 kinf. This process also depends on the use of burnable absorbers, which disrupt the first assumption about the constantly decreasing assembly kinf.

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.

flow rate decreaseIn 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

Xenon - 135. Neutron absorption and scattering. Comparison of cross-sections.

Xenon – 135. Neutron absorption and scattering. Comparison of cross-sections.
Source: JANIS (Java-based Nuclear Data Information Software); The JEFF-3.1.1 Nuclear Data Library

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 (FQ(z) limit) is highly restrictive, large xenon spatial oscillations have to be minimized to prevent exceeding FQ 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 = Itop – Ibottom

where Itop and Ibottom are expressed as a fraction of rated thermal power.

 

Effect of Thermal Power

axial-temperature-profile

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.

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

reactor power - 75 to 100 of rated power

Power increase. 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 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 = Itop – Ibottom

where Itop and Ibottom are expressed as a fraction of rated thermal power.

Nuclear and Reactor Physics:

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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.
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  4. E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.

See above: