Reactor Kinetics

Reactor kinetics is the study of the time-dependence of the neutron flux for postulated changes in the macroscopic cross-sections. It is also referred to as reactor kinetics without feedback.

To study the kinetic behavior of the system, engineers usually use point kinetics equations. Although the number of delayed neutrons per fission neutron is quite small (typically below 1%) and thus does not contribute significantly to the power generation, they play a crucial role in reactor control. They are essential from the point of view of reactor kinetics and reactor safety.

In this section, we will study the time-dependent behavior of nuclear reactors. Understanding the time-dependent behavior of the neutron population in a nuclear reactor in response to either a planned change in the reactivity of the reactor or to unplanned and abnormal conditions is the most important in nuclear reactor safety.

Nuclear reactor kinetics deals with transient neutron flux changes resulting from a departure from the critical state, from some reactivity insertion. Such situations arise during operational changes such as control rods motion, environmental changes such as a change in boron concentration, or accidental disturbances in the reactor steady-state operation.

In general:

• Reactor Kinetics. Reactor kinetics is the study of the time-dependence of the neutron flux for postulated changes in the macroscopic cross-sections. It is also referred to as reactor kinetics without feedback.
• Reactor Dynamics. Reactor dynamics study the time-dependence of the neutron flux when the macroscopic cross-sections are allowed to depend in turn on the neutron flux level. It is also referred to as reactor kinetics with feedback and spatial effects.

The time-dependent behavior of nuclear reactors can also be classified by the time scale as:

• Short-term kinetics describes phenomena that occur over times shorter than a few seconds. This comprises the response of a reactor to either a planned change in the reactivity or to unplanned and abnormal conditions. In this section, we will introduce especially point kinetics equations.
• Medium-term kinetics describes phenomena that occur for several hours to a few days. This comprises especially effects of neutron poisons on the reactivity (i.e., Xenon poisoning or spatial oscillations).
• Long-term kinetics describes phenomena that occur over months or even years. This comprises all long-term changes in fuel composition due to fuel burnup.

This chapter is concerned especially with short-term kinetics and the point kinetics equations. At first, have to start with an introduction to prompt and delayed neutrons because they play an important role in short-term reactor kinetics. Although the number of delayed neutrons per fission neutron is quite small (typically below 1%) and thus does not contribute significantly to the power generation, they play a crucial role in reactor control. They are essential from the point of view of reactor kinetics and reactor safety. Their presence completely changes the dynamic time response of a reactor to some reactivity change, making it controllable by control systems such as the control rods.

Delayed neutrons allow to operate a reactor in a prompt subcritical, delayed critical condition. All power reactors are designed to operate in delayed critical conditions and are provided with safety systems to prevent them from ever achieving prompt criticality.

Simple Point Kinetics Equation

As seen in previous chapters, neutrons are multiplied by a factor keff from one neutron generation to the next. Therefore, the multiplication environment (nuclear reactor) behaves like an exponential system, which means the power increase is not linear but exponential.

The effective multiplication factor in a multiplying system measures the change in the fission neutron population from one neutron generation to the subsequent generation.

• keff < 1. Suppose the multiplication factor for a multiplying system is less than 1.0. In that case, the number of neutrons decreases 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 exponentially increases in time (with the mean generation time). This condition is known as the supercritical state.

But we have not yet discussed the duration of a neutron generation, which means how many times in one second we have to multiply the neutron population by a factor keff. This time determines the speed of the exponential growth. But as was written, there are different types of neutrons: prompt neutrons and delayed neutrons, which completely change the kinetic behavior of the system. Therefore such a discussion will be not trivial.

To study the kinetic behavior of the system, engineers usually use point kinetics equations. The name point kinetics is used because, in this simplified formalism, the neutron flux shape and the neutron density distribution are ignored. The reactor is therefore reduced to a point. The following section will introduce point kinetics, starting with point kinetics in its simplest form.

Derivation of Simple Point Kinetics Equation

Let n(t) be the number of neutrons as a function of time t and l the prompt neutron lifetime, which is the average time from a prompt neutron emission to either its absorption (fission or radiative capture) or its escape from the system. The average number of neutrons that disappear during a unit time interval dt is n.dt/l. But each disappearance of a neutron contributes an average of k new neutrons.

Finally, the change in the number of neutrons during a unit time interval dt is:

where:

n(t) = transient reactor power

n(0) = initial reactor power

τe = reactor period

The reactor period, τe, or e-folding time, is defined as the time required for the neutron density to change by a factor e = 2.718. The reactor period is usually expressed in units of seconds or minutes. The smaller the value of τe, the more rapid the change in reactor power. The reactor period may be positive or negative.

Simple Point Kinetics Equation without Delayed Neutrons

An equation governing the neutron kinetics of the system without source and with the absence of delayed neutrons is the point kinetics equation (in a certain form). 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 prompt neutron lifetime. The role of prompt neutron lifetime is evident, and shorter lifetimes give simply faster responses to 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 form of point kinetics equation (without source and delayed neutrons).

This simple point kinetics equation is often expressed in terms of reactivity and prompt generation time, Λ, as:

where

• ρ = (k-1)/k is the reactivity, which describes the deviation of an effective multiplication factor from unity.
• Λ = l/keff = prompt neutron generation time, the average time from a prompt neutron emission to absorption that results only in fission.

Both forms of the point kinetics equation are valid. The equation using Λ, prompt neutron generation time, is usually better for calculations. This is because most reactivity transients are induced by changes in the absorption cross-section rather than in the fission cross-section. The prompt neutron lifetime is not constant during these transients, whereas the prompt generation time remains constant.

Example:

Let us consider that the prompt neutron lifetime is ~2 x 10-5, and k (k – neutron multiplication factor) will be 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 = l / (k– 1) = 2 x 10-5 / (1.0001 – 1) = 0.2s

This is a very short period. In one second, the neutron flux (and power) in the reactor would increase by a factor of e5 = 2.7185. In 10 seconds, the reactor would pass through 50 periods, and the power would increase by e50.

Furthermore, in the case of fast reactors in which prompt neutron lifetimes are of the order of 10-7 seconds, the response of such a small reactivity insertion will be even more unimaginable. In the case of 10-7, the period will be:

T = l / (k– 1) = 10-7 / (1.0001 – 1) = 0.001s

Reactors with such kinetics would be very difficult to control. Fortunately, this behavior is not observed in any multiplying system. Actual reactor periods are observed to be considerably longer than computed above, and therefore the nuclear chain reaction can be controlled more easily. The longer periods are observed due to the presence of the delayed neutrons.

Prompt neutron lifetime, l, is the average time from a prompt neutron emission to either its absorption (fission or radiative capture) or its escape from the system. This parameter is defined in multiplying or also in non-multiplying systems. In both systems, the prompt neutron lifetimes depend strongly on:

• the material composition of the system
• multiplying – non-multiplying system
• the system with or without thermalization
• isotopic composition of the system
• the geometric configuration of the system
• homogeneous or heterogeneous system
• the shape of the entire system
• size of the system

In an infinite reactor (without escape), prompt neutron lifetime is the sum of the slowing downtime and the diffusion time.

l=ts + td

In an infinite thermal reactor ts << td and therefore l ≈ td. The typical prompt neutron lifetime in thermal reactors is on the order of 10−4 seconds. Generally, the longer neutron lifetimes occur in systems where the neutrons must be thermalized to be absorbed.

Systems in which most of the neutrons are absorbed in higher energies and the neutron thermalization is suppressed (e.g., in fast reactors) have much shorter prompt neutron lifetimes. The typical prompt neutron lifetime in fast reactors is on the order of 10−7 seconds.

Prompt Generation Time - Mean Generation Time

In multiplying systems, the absorption of a prompt fission neutron can initiate a fission reaction, l is equal to the average time between two generations of prompt neutrons (at keff=1). This time is known as the prompt neutron generation time.

Prompt Neutron Generation Time (or Mean Generation Time), Λ, is the average time from a prompt neutron emission to a capture that results only in fission. The prompt neutron generation time is designated as:

Λ = l/keff

The prompt generation time changes with the fuel burnup in power reactors, and LWRs increase with the fuel burnup. It is simple, and fresh uranium fuel contains much fissile material (in the case of uranium fuel, about 4% of 235U). This causes significant excess of reactivity, and this excess must be compensated via chemical shim (in case of PWRs) or burnable absorbers.

Due to these factors (high probability of absorption in fuel and high probability of absorption in moderator), the prompt neutron lives much shorter, and the prompt neutron lifetime is low. With fuel burnup, the amount of fissile material and the absorption in the moderator decreases, and therefore the prompt neutron can “live” much longer.

Effect of Prompt Neutron Lifetime on Nuclear Safety
The prompt neutron lifetime belongs to key neutron-physical characteristics of reactor core. Its value depends especially on the type of the moderator and the energy of the neutrons causing fission. Its importance for nuclear reactor safety has been well known for a long time.

The longer prompt neutron lifetimes can substantially improve the kinetic response of the reactor (the longer prompt neutron lifetime gives simply a slower power increase). For example, under RIA conditions (Reactivity-Initiated Accidents), reactors should withstand a jump-like insertion of relatively large (~1 \$ or even more) positive reactivity, and the PNL (prompt neutron lifetime) plays here the key role. Therefore the PNL should be verified in a reload safety evaluation process.

In some cases (especially in some fast reactors), reactor cores can be modified to increase the PNL and improve nuclear safety.

Simple Point Kinetics Equation with Delayed Neutrons

The simplest equation governing the neutron kinetics of the system with delayed neutrons is the simple point kinetics equation with delayed neutrons. 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).

ld = (1 – β).lp + ∑li . βi => ld = (1 – β).lp + ∑τi . βi

where

• (1 – β) is the fraction of all neutrons emitted as prompt neutrons
• lp is the prompt neutron lifetime
• τi is the mean precursor lifetime, the inverse value of the decay constant τi = 1/λi
• The weighted delayed generation time is given by τ = ∑τi . βi / β = 13.05 s
• Therefore the weighted decay constant λ = 1 / τ ≈ 0.08 s-1

The number, 0.08 s-1, is relatively high and has a dominating effect on reactor time response, although delayed neutrons are a small fraction of all neutrons in the core. This is best illustrated by calculating a weighted mean generation time with delayed neutrons:

ld = (1 – β).lp + ∑τi . βi = (1 – 0.0065). 2 x 10-5 + 0.085 = 0.00001987 + 0.085 ≈ 0.085

In short, the mean generation time with delayed neutrons is about ~0.1 s, rather than ~10-5 as in section Prompt Neutron Lifetime, where the delayed neutrons were omitted.

The role of ld is evident, and longer lifetimes give simply slower responses to multiplying systems. The role of reactivity (keff – 1) is also evident, and higher reactivity gives the simply larger response of the multiplying system.

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):

Example:

Let us consider that the mean generation time with delayed neutrons is ~0.085, and k (k – neutron multiplication factor) will increase 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 the case of LWRs (e.g., one step by control rods). 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 power) in the reactor would increase by a factor of e = 2.718. This is a completely different dimension of the response on reactivity insertion than the case without delayed neutrons, where the reactor period was 1 second.

Six Groups of Delayed Neutrons

Reactor-kinetic calculations considering many initial conditions would be correct, but they would also be very complicated. Therefore G. R. Keepin and his co-workers suggested to group together the precursors based on their half-lives. Therefore delayed neutrons are traditionally represented by six delayed neutron groups, whose yields and decay constants (λ) are obtained from nonlinear least-squares fits experimental measurements. This model has the following disadvantages:

• All constants for each group of precursors are empirical fits to the data.
• They cannot be matched with decay constants of specific precursors.

Although this six-group parameterization still satisfies the requirements of commercial organizations, higher accuracy of the delayed neutron yields and a better energy resolution in the delayed neutron spectra is desired.

It was recognized that the half-lives in the six-group structure do not accurately reproduce the asymptotic die-away time constants associated with the three longest-lived dominant precursors: 87Br, 137I, and 88Br.

This model may be insufficient, especially in the case of epithermal reactors, because virtually all delayed neutron activity measurements have been performed for fast or thermal-neutron-induced fission. In the case of fast reactors, the nuclear fission of six fissionable isotopes of uranium and plutonium is important, and the accuracy and energy resolution may play an important role.

Photoneutrons
In nuclear reactors the gamma radiation plays a significant role also in reactor kinetics and in a subcriticality control. Especially in nuclear reactors with D2O moderator (CANDU reactors) or Be reflectors (some experimental reactors). Neutrons can also be produced in (γ, n) reactions, and therefore they are usually referred to as photoneutrons.

Under certain conditions, a high-energy photon (gamma-ray) can eject a neutron from a nucleus. It occurs when its energy exceeds the binding energy of the neutron in the nucleus. Most nuclei have binding energies over 6 MeV, above the energy of most gamma rays from fission. On the other hand, few nuclei with sufficiently low binding energy are of practical interest. These are: 2D, 9Be6Li, 7Li, and 13C. As can be seen from the table, the lowest threshold have 9Be with 1.666 MeV and 2D with 2.226 MeV.

In the case of deuterium, neutrons can be produced by the interaction of gamma rays (with a minimum energy of 2.22 MeV) with deuterium:

Because gamma rays can be emitted by fission products with certain delays, and the process is very similar to that through which a “true” delayed neutron is emitted, photoneutrons are usually treated no differently than regularly delayed neutrons in the kinetic calculations. Photoneutron precursors can also be grouped by their decay constant, similarly to “real” precursors. The table below shows the relative importance of source neutrons in CANDU reactors by showing the makeup of the full power flux.

Although photoneutrons are important, especially in CANDU reactors, deuterium nuclei are always present (~0.0156%) in the light water of LWRs. Moreover, the capture of neutrons in the hydrogen nucleus of the water molecules in the moderator yields small amounts of D2O, enhancing the heavy water concentration. Therefore, in LWRs kinetic calculations, photoneutrons from D2O are treated as additional groups of delayed neutrons with characteristic decay constants λj and effective group fractions.

After a nuclear reactor has been operating at full power for some time, there will be a considerable build-up of gamma rays from the fission products. This high gamma flux from short-lived fission products will decrease rapidly after shutdown. The photoneutron source decreases with the decay of long-lived fission products that produce delayed high-energy gamma rays in the long term. The photoneutron source drops slowly, decreasing a little each day. The longest-lived fission product with gamma-ray energy above the threshold is 140Ba with a half-life of 12.75 days.

The amount of fission products present in the fuel elements depends on how long the reactor has been operated before shutdown and at which power level has been operated before shutdown. Photoneutrons are usually a major source in a reactor and ensure sufficient neutron flux on source range detectors when the reactor is subcritical in long-term shutdown.

Compared with fission neutrons, which make a self-sustaining chain reaction possibledelayed neutrons make reactor control possible, and photoneutrons are important at low power operation.

Effective Delayed Neutron Fraction – βeff

The delayed neutron fraction, β, is the fraction of delayed neutrons in the core at creation at high energies. But in the case of thermal reactors, the fission can be initiated mainly by a thermal neutron. Thermal neutrons are of practical interest in the study of thermal reactor behavior. The effective delayed neutron fraction usually referred to as βeff, is the same fraction at thermal energies.

The effective delayed neutron fraction reflects the ability of the reactor to thermalize and utilize each neutron produced. The β is not the same as the βeff due to the fact delayed neutrons do not have the same properties as prompt neutrons released directly from fission. In general, delayed neutrons have lower energies than prompt neutrons. Prompt neutrons have initial energy between 1 MeV and 10 MeV, with an average energy of 2 MeV. Delayed neutrons have initial energy between 0.3 and 0.9 MeV with an average energy of 0.4 MeV.

Therefore, a delayed neutron traverses a smaller energy range in thermal reactors to become thermal. It is also less likely to be lost by leakage or parasitic absorption than the 2 MeV prompt neutron. On the other hand, delayed neutrons are also less likely to cause fast fission because their average energy is less than the minimum required for fast fission to occur.

These two effects (lower fast fission factor and higher fast non-leakage probability for delayed neutrons) tend to counteract each other and form a term called the importance factor (I). The importance factor relates the average delayed neutron fraction to the effective delayed neutron fraction. As a result, the effective delayed neutron fraction is the product of the average delayed neutron fraction and the importance factor.

βeff = β . I

The delayed and prompt neutrons have a difference in their effectiveness in producing a subsequent fission event. Since the energy distribution of the delayed neutrons also differs from group to group, the different groups of delayed neutrons will also have different effectiveness. Moreover, a nuclear reactor contains a mixture of fissionable isotopes. Therefore, the importance factor is insufficient in some cases, and an importance function must be defined.

For example:

In a small thermal reactor with highly enriched fuel, the increase in fast non-leakage probability will dominate the decrease in the fast fission factor, and the importance factor will be greater than one.

In a large thermal reactor with low enriched fuel, the decrease in the fast fission factor will dominate the increase in the fast non-leakage probability, and the importance factor will be less than one (about 0.97 for a commercial PWR).

In large fast reactors, the decrease in the fast fission factor will also dominate the increase in the fast non-leakage probability, and the βeff is less than β by about 10%.

Mean Generation Time with Delayed Neutrons
Mean Generation Time with Delayed Neutrons, ld, is the weighted average of the prompt generation times and a delayed neutron generation time. The delayed neutron generation time, τ, is the weighted average of mean precursor lifetimes of the six groups (or more groups) of delayed neutron precursors.

It must be noted the true lifetime of delayed neutrons (the slowing downtime and the diffusion time) is very short compared with the mean lifetime of their precursors (ts + td << τi). Therefore τi is also equal to the mean lifetime of a neutron from the ith group, that is, τi = li, and the equation for mean generation time with delayed neutrons is the following:

ld = (1 – β).lp + ∑li . βi => ld = (1 – β).lp + ∑τi . βi

where

• (1 – β) is the fraction of all neutrons emitted as prompt neutrons
• lp is the prompt neutron lifetime
• τis the mean precursor lifetime, the inverse value of the decay constant τi = 1/λi
• The weighted delayed generation time is given by τ = ∑τi . βi / β = 13.05 s
• Therefore the weighted decay constant λ = 1 / τ ≈ 0.08 s-1

The number, 0.08 s-1, is relatively high and has a dominating effect on reactor time response, although delayed neutrons are a small fraction of all neutrons in the core. This is best illustrated by calculating a weighted mean generation time with delayed neutrons:

ld = (1 – β).lp + ∑τi . βi = (1 – 0.0065). 2 x 10-5 + 0.085 = 0.00001987 + 0.085 ≈ 0.085

In short, the mean generation time with delayed neutrons is about ~0.1 s, rather than ~10-5 as in section Prompt Neutron Lifetime, where the delayed neutrons were omitted.

Effect of Delayed Neutrons on Reactor Control
Despite the fact the number of delayed neutrons per fission neutron is quite small (typically below 1%) and thus does not contribute significantly to the power generation, they play a crucial role in the reactor control and are essential from the point of view of reactor kinetics and reactor safety. Their presence completely changes the dynamic time response of a reactor to some reactivity change, making it controllable by control systems such as the control rods.

Delayed neutrons allow to operate a reactor in a prompt subcritical, delayed critical condition. All power reactors are designed to operate in delayed critical conditions and are provided with safety systems to prevent them from ever achieving prompt criticality.

For typical PWRs, the prompt criticality occurs after positive reactivity insertion of βeff(i.e., keff ≈ 1.006 or ρ = +600 pcm). In power reactors, such a reactivity insertion is practically impossible to insert (in case of normal and abnormal operation), especially when a reactor is in power operation mode and a reactivity insertion causes heating of a reactor core. Due to the presence of reactivity feedbacks, the positive reactivity insertion is counterbalanced by the negative reactivity from moderator and fuel temperature coefficients. The presence of delayed neutrons is of importance also from this point of view because they provide time also to reactivity feedbacks to react on undesirable reactivity insertion.

Interactive chart – Infinite Multiplying System Without Source and Delayed Neutrons
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).

Point Kinetics Equations

Both previous simple point kinetics equations are only an approximation because they use many simplifications. The simple point kinetics equation with delayed neutrons completely fails for higher reactivity insertions, where is a significant difference between the production of prompt and delayed neutrons. Therefore a more accurate model is required. The exact point kinetics equations that can be derived from the general neutron balance equations without making any approximations are:

In the equation for neutrons, the first term on the right-hand side is the production of prompt neutrons in the present generation, k(1-β)n/l, minus the total number of neutrons in the preceding generation, -n/l. The second term is the production of delayed neutrons in the present generation. As can be seen, the rate of absorption of neutrons is the same as in the simple model (-n/l). But a distinction is between the direct channel for prompt neutrons (1-β) production and the delayed channel resulting from radioactive decay of precursor nuclei (λiCi).

In the equation for precursors, there is a balance between the production of the precursors of i-th group and their decay after the decay constant λi. As can be seen, the decay rate of precursors is the radioactivity rate (λiCi). The production rate is proportional to the number of neutrons times βi, which is defined as the fraction of the neutrons that appear as delayed neutrons in the ith group.

As can be seen, the point kinetics equations include two differential equations, one for the neutron density n(t) and the other for precursors concentration C(t).

Again, the point kinetics equations are often expressed in terms of reactivity (ρ = (k-1)/k) and prompt generation time, Λ, as:

Both forms of the point kinetics equation are valid. The equation using Λ, prompt neutron generation time, is usually better for calculations. This is because most reactivity transients are induced by changes in the absorption cross-section rather than in the fission cross-section. The prompt neutron lifetime is not constant during these transients, whereas the prompt generation time remains constant.

The previous equation defines the reactivity of a reactor, which describes the deviation of an effective multiplication factor from unity. For critical conditions, the reactivity is equal to zero. The larger the absolute value of reactivity in the reactor core, the further the reactor is from criticality. The reactivity may be used to measure a reactor’s relative departure from criticality. According to the reactivity, we can classify the different reactor states and the related consequences as follows:

prompt critical state
The prompt critical state is defined as:

keff > 1; ρ ≥ βeff, where the reactivity of a reactor is higher than the effective delayed neutron fraction. In this case, the production of prompt neutrons alone is enough to balance neutron losses and increase the neutron population. The number of neutrons is exponentially increasing in time (as rapidly as the prompt neutron generation lifetime ~10-5s).

delayed supercritical state
The prompt subcritical and delayed supercritical state is defined as:

keff > 1; 0 < ρ < βeff, where the reactivity of a reactor is higher than zero and lower than the effective delayed neutron fraction. In this case, the production of prompt neutrons alone is insufficient to balance neutron losses, and the delayed neutrons are needed to sustain the chain reaction. The neutron population increases, but much more slowly (as the mean generation lifetime with delayed neutrons ~0.1 s).

delayed critical
The prompt subcritical and delayed critical state is defined as:

keff = 1; ρ = 0, where the reactivity of a reactor is equal to zero. In this case, the production of prompt neutrons alone is insufficient to balance neutron losses, and the delayed neutrons are needed to sustain the chain reaction. There is no change in neutron population in time, and the chain reaction will be self-sustaining. This state is the same as the critical state from basic classification.

delayed subcritical state
The prompt subcritical and delayed subcritical state is defined as:

keff < 1; ρ < 0, where the reactivity of a reactor is lower than zero. In this case, the production of all neutrons is insufficient to balance neutron losses, and the chain reaction is not self-sustaining. If the reactor core contains external or internal neutron sources, the reactor is in the state that is usually referred to as the subcritical multiplication.

Inhour Equation

If the reactivity is constant, the model of point kinetics equations contains a set (1 + 6) of linear ordinary differential equations with constant coefficient and can be solved analytically. Solution of six-group point kinetics equations with Laplace transformation leads to the relation between the reactivity and the reactor period. This relation is known as the inhour equation (which comes from the inverse hour, when used as a unit of reactivity that corresponded to e-fold neutron density change during one hour) may be derived.

General Form:

The point kinetics equations may be solved for the case of an initially critical reactor without an external source in which the properties are changed at t = 0 in such a way as to introduce a step reactivity ρ0 which is then constant over time. The system of coupled first-order differential equations can be solved with Laplace transformation or by trying the solution n(t) = A.exp(s.t) (equation for the neutron flux) and Ci(t) = Ci,0.exp(s.t) (equations for the density of precursors).

Substitution of these assumed exponential solutions in the equation for precursors gives the relation between the coefficients of the neutron density and the precursors.

The subsequent substitution in the equation for neutron density yields an equation for s, which after some manipulation can be written as:

This equation is known as the inhour equation since the constants of s0 – 6 were originally determined in inverse hours. For a given value of the reactivity ρ, the associated values of s0 – 6 are determined with this equation. The following figure shows the relation between ρ and roots s graphically. From this figure, it can be seen that for a given value of ρ, seven solutions exist for s. The figure indicates that for positive reactivity, only s0 is positive. The remaining terms rapidly die away, yielding an asymptotic solution in the form:

where s0 = 1/τe is the reactor’s stable reactor period or asymptotic period. This root, s0, is positive for ρ > 0 and negative for ρ < 0. Therefore this root describes the reactor response, lasting after the transition phenomena have died out. The figure also shows that a negative reactivity leads to a negative period: All si is negative, but the root s0 will die away more slowly than the others. Thus the solution n(t) = A0exp(s0t) is valid for positive as well as negative reactivity insertions.

A plot of ρ vs. τe must be constructed using the delayed neutron data for a particular fissionable isotope or a mix of isotopes and for a given prompt generation time to determine the reactivity required to produce a given period. It is convenient to use the following inhour equation to determine the stable reactor period, which results from a given reactivity insertion.

Reactor Period Form of Inhour Equation

where:

τe = reactor period

The first term in this formula is the prompt term, and it causes the positive reactivity insertion to be followed immediately by an immediate power increase called the prompt jump. This power increase occurs because the production rate of prompt neutrons changes immediately as the reactivity is inserted. After the prompt jump, the rate of change of power cannot increase any more rapidly than the built-in time delay the precursor half-lives allow. Therefore the second term in this formula is called the delayed term. The presence of delayed neutrons causes the power rise to be controllable, and the reactor can be controlled by control rods or another reactivity control mechanism.

Reactor Period and Startup Rate
Reactivity is not directly measurable, and therefore most power reactors procedures do not refer to it, and most technical specifications do not limit it. Instead, they specify a limiting rate of neutron power rise (measured by excore detectors), commonly called a startup rate (especially in the case of PWRs).

The reactor startup rate is defined as the number of factors often that power changes in one minute. Therefore the units of SUR are powers of ten per minute or decades per minute (dpm). The relationship between reactor power and startup rate is given by the following equation:

n(t) = n(0).10SUR.t

where:
SUR = reactor startup rate [dpm – decades per minute]
t = time during reactor transient [minute]

The higher the value of SUR, the more rapid the change in reactor power. The startup rate may be positive or negative. If SUR is positive, reactor power increases, and if SUR is negative, reactor power decreases. The relationship between reactor period and startup rate is given by the following equations:

Example:

Suppose keff = 1.0005 in a reactor with a generation time ld = 0.01s. For this state calculate the reactor period – τe, doubling time – DT and the startup rate (SUR).
ρ = 1.0005 – 1 / 1.0005 = 50 pcm

τe = ld / k-1 = 0.1 / 0.0005 = 200 s

DT = τe . ln2 = 139 s

SUR = 26.06 / 200 = 0.13 dpm

Special Cases of Inhour Equation

Spoiler title
Assume that ρ is very small. This causes the first root, s0, of the reactivity to be also very small. In this case, s0” λi and therefore, we can ignore the term s0 in the denominator of the inhour equation. The inhour equation then takes the form:

As can be seen, this special case results in the same period as in the case of the simple point kinetics equation, which also uses the mean generation time with delayed neutrons (ld):

Thus for small reactivities—positive or negative—, the reactor period is governed almost completely by the delayed neutron properties. Although the amount of delayed neutrons is only on the order of tenths of a percent of the total amount, the timescale in seconds (τi) plays an extremely important role.

The assumption is s0λi. The largest value of τi = 1/λi = 80s, this formula is valid for periods, τe, higher than 80s. On the other hand, this formula is useful and accurate enough for most purposes for reactivities up to about ρ = 0.0005 = 50pcm.

Note that:

ld = (1 – β).lp + ∑li . βi => ld = (1 – β).lp + ∑τi . βi

where

• (1 – β) is the fraction of all neutrons emitted as prompt neutrons
• lp is the prompt neutron lifetime
• τi is the mean precursor lifetime, the inverse value of the decay constant τi = 1/λi
• The weighted delayed generation time is given by τ = ∑τi . βi / β = 13.05 s
• Therefore the weighted decay constant λ = 1 / τ ≈ 0.08 s-1

The number, 0.08 s-1, is relatively high and has a dominating effect on reactor time response, although delayed neutrons are a small fraction of all neutrons in the core. This is best illustrated by calculating a weighted mean generation time with delayed neutrons:

ld = (1 – β).lp + ∑τi . βi = (1 – 0.0065). 2 x 10-5 + 0.085 = 0.00001987 + 0.085 ≈ 0.085

In short, the mean generation time with delayed neutrons is about ~0.1 s, rather than ~10-5 as in section Prompt Neutron Lifetime, where the delayed neutrons were omitted.

Example:

Let us consider that the mean generation time with delayed neutrons is ~0.085, and k (k – 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 the case of LWRs (e.g., one step by control rods). 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 power) in the reactor would increase by a factor of e = 2.718. This is a completely different dimension of the response on reactivity insertion in comparison with the case without the presence of delayed neutrons, where the reactor period was 1 second.

ρ ≥ β - high reactivity - prompt criticality
Assume that ρ is higher than β.  In this case, the production of prompt neutrons alone is enough to balance neutron losses and increase the neutron population. The number of neutrons is exponentially increasing in time (as rapidly as the prompt neutron generation lifetime ~10-5s).

This causes the first root, s0, of the reactivity, to become larger than each λi. In this case, λis0 and therefore, we can ignore the term λi in the denominator of the inhour equation. The inhour equation then takes the form:

As can be seen, this special case results in the same period as in the case of simple point kinetics equation without delayed neutrons.

Reaching the reactivity of β (e.g., 650pcm) completely changes the response of a reactor; therefore, this situation can only be accidental. As prompt criticality is reached, the distinction between the prompt jump and the reactor period vanishes. The prompt neutron lifetime rather than the delayed neutron half-lives largely determine the rate of exponential increase. Reactors with such kinetics would be very difficult to control by mechanical means such as the movement of control rods. In normal operation, the reactivity of a reactor must remain far below the prompt criticality threshold with sufficient margin.

In design basis accidents (DBA), such kinetics can occur. For example, under RIA conditions (Reactivity-Initiated Accidents), reactors should withstand a jump-like insertion of relatively large (~1 \$ or even more) positive reactivity (e.g., in case of control rod ejection), and the PNL (prompt neutron lifetime) plays here the key role. The longer prompt neutron lifetimes can substantially improve the kinetic response of the reactor (the longer prompt neutron lifetime gives simply a slower power increase). Therefore the PNL should be verified in a reload safety evaluation (RSE) process. Management of this accident is based on reactivity feedback, especially on the doppler temperature coefficient (DTC). This coefficient is of the highest importance in reactor stability. The doppler temperature coefficient is generally considered even more important than the moderator temperature coefficient (MTC). Especially in the case of a control rod ejection, the doppler temperature coefficient will be the first and the most important feedback that will compensate for the inserted positive reactivity. The time for heat to be transferred to the moderator is usually measured in seconds, while the fuel temperature coefficient is effective almost instantaneously. Therefore this coefficient is also called the prompt temperature coefficient because it causes an immediate response to changes in fuel temperature.

ρ < 0 - smallest negative period possible
Assume that ρ is negative and ρ β. This case is typical for the emergency shutdown of a reactor. This event is also called a reactor trip or SCRAM. A reactor “SCRAM” (or “trip”) is the rapid insertion or fall of the control rods into the core to stop the fission chain reaction. In PWRs, all control rods are usually inserted within two to four seconds. In this case, the reactivity of the multiplying system becomes quickly negative and higher than β (in absolute values).

But even if it were possible to insert an infinite negative reactivity, the neutron flux would not immediately fall to zero. Prompt neutrons will be absorbed almost immediately. It is consistent with the prompt drop formula. Therefore the resulting neutron flux will be:

Obviously, the neutron flux cannot drop below the value βn1. The real values are much higher. The integral worth of all control and emergency rods (PWRs) is, for example -9000pcm. It is equal to ρ = -9000/600 = -15β = -0.09 (β= 600pcm = 0.006)

For this negative reactivity, the prompt drop is equal to:

n2/n1 = 0.006/(0.006+0.09)=0.063

which is about ten times higher than in the case of an infinite negative reactivity insertion.

The neutron flux then continues to fall according to the stable period. The first root of the reactivity equation occurs at s0 = – λ1. The decay is constant of the long-lived precursor’s group. The shortest negative stable period is then τe = – 1/λ1 = -80s. The neutron flux cannot be reduced more rapidly than this period. On the other hand, the prompt drop causes an immediate drop to about 6% of rated power. Within a few tens of seconds, the thermal power that originates from nuclear fission is below the thermal power that originates from decay heat.

Reactivity Pulse – Impulse Characteristics

We will now study the response of a reactor on a reactivity pulse, which is represented by the Dirac delta function, δ(t). Strictly speaking, the Dirac delta function is not a function, but a so-called distribution, but here the function form will be used, in which the delta function is defined as follows:

the reactivity pulse can be mathematically expressed as ρ(t) = ρ0 . δ(t). Using the inverse Laplace transformation and the system transfer function, G(s), it can be derived that the pulse reactivity insertion causes a transient which is characterized by the following relations:

That means the prompt neutron lifetime plays a key role in the first part of the transient, while the delayed neutrons play a key role in the steady-state neutron level.

Oscillation of Reactivity – Frequency Characteristics

We will now study the response of a reactor on a reactivity oscillation, which is represented by the following function: ρ(t) = ρ0 . cos(ωt). Where ρ0 is the amplitude of the input signal (forcing function), and ω is the signal frequency expressed in radians per second.

Using the inverse Laplace transformation and the system transfer function, G(s), it can be derived that the system response is strongly dependent on the frequency, ω.

Approximate Solution of Point Kinetics Equations

Sometimes, it is convenient to predict qualitatively the behavior of a reactor. The exact solution can be obtained relatively easily using computers. Especially for illustration, the following approximations are discussed in the following sections:

• Prompt Jump Approximation
• Prompt Jump Approximation with One Group of Delayed Neutrons
• Constant Delayed Neutron Source Approximation
Prompt Jump Approximation - Prompt Drop
As can be seen from the solution of exact point kinetics equation, any reactivity insertion (ρ < β) causes at first a sharp change in prompt neutrons population, and then the neutron response is slowed as a result of the more slowly changing number of delayed neutrons. The rapid response is a result of the small value of prompt neutron generation time in the denominator of the point kinetics equation.

Suppose we are interested in long-term behavior (asymptotic period) and not interested in the details of the prompt jump. In that case, we can simplify the point kinetics equations by assuming that the prompt jump takes place instantaneously in response to any reactivity change. This approximation is known as the Prompt Jump Approximation (PJA). Due to prompt neutrons, the rapid power change is neglected, corresponding to taking dn/dt |0 = 0 in the point kinetics equations. That means the point kinetics equations are as follows:

From the equation for neutron flux and the assumption that the delayed neutron precursor population does not respond instantaneously to a change in reactivity (i.e., Ci,1 = Ci,2), it can be derived that the ratio of the neutron population just after and before the reactivity change is equal to:

The prompt-jump approximation is usually valid for smaller reactivity insertion, for example, for ρ < 0.5β. It is usually used with another simplification and the one delayed precursor group approximation.

Prompt Jump Approximation with One Group of Delayed Neutrons
In the previous section, we have simplified the point kinetics equation using prompt jump approximation (PJA). This eliminated the fast time scale due to prompt neutrons. This section considers that delayed neutrons are produced only by one group of precursors with the same decay constant (averaged) and delayed neutron fraction. Point kinetics equation using PJA and one group of delayed neutrons becomes:

Assuming that the reactivity is constant and n1/n0 can be determined from the prompt jump formula, this equation leads to a very simple formula:

Constant Delayed Neutron Source Approximation
As can be seen from the solution of exact point kinetics equation, any reactivity insertion (ρ < β) causes at first a sharp change in prompt neutrons population, and then the neutron response is slowed as a result of the more slowly changing number of delayed neutrons. The rapid response is a result of the small value of prompt neutron generation time in the denominator of the point kinetics equation.

Suppose we are interested in short-term behavior and not in the details of the asymptotic behavior. In that case, we can simplify the point kinetics equations by assuming that the production of the delayed neutrons is constant and equal to the production at the beginning of the transient. This approximation is known as the Constant Delayed Neutron Source Approximation (CDS), in which changes in the number of delayed neutrons are neglected, corresponding to taking dCi(t)/dt = 0 and Ci(t) = Ci,0 in the point kinetics equations. That means the point kinetics equations are as follows:

The above equation can be solved analytically, and assuming that the reactivity is constant, the solution is given as:

Experimental Methods of Reactivity Determination

There are two main experimental methods for fundamental reactor physics measurements: kinetic and static.

• Static methods are used to determine time-independent core characteristics. These methods can be used to describe phenomena that occur independently of time, and on the other hand, they cannot be used to determine the most dynamic characteristics.
• Kinetic methods are used to study parameters (parameters of delayed neutrons etc.) that determine short-term and medium-term kinetics.

There are three main kinetic methods for the experimental determination of neutron kinetics parameters:

Measurement of Asymptotic Period
The asymptotic period measurement is based on measurement of the exponential rise of neutron flux level during step increase in reactivity. In general, the stable reactor period (or asymptotic period), τe, is defined as the time required for the neutron density to change by a factor e = 2.718.

As was written in previous chapters, we can expect that the solution of point kinetics equation can be n(t) = A.exp(s.t) and Ci(t) = Ci,0.exp(s.t). In the asymptotic term (i.e., after the transition phenomena have died out), the asymptotic solution is in the form:

where s0 = 1/e is the stable reactor period or asymptotic period of the reactor. This root, s0, is positive for ρ > 0 and negative for ρ < 0. Therefore this root describes the reactor response, lasting after the transition phenomena have died out. The measurement of the asymptotic period can be used to determine (directly from inhour equation) the reactivity inserted into the system.

Rod Drop Method
The rod drop method belongs to a group of reactivity perturbation methods. This method is based on the study of the transient response of the reactor to a rapid insertion of high negative reactivity. This rapid reactivity insertion is usually performed by dropping the reactor control rods when the reactor is in a critical state. In this case, the reactivity inserted can be determined from the measurement of the prompt drop.

As was described in the Prompt Jump Approximation, the response of a neutron detector (n1 ➝ n2) immediately after a control rod is dropped into a critical reactor (ρ1 = 0) is related by:

which allows the determination of the reactivity worth of the rod. The rod drop method is advantageous because it is very quick to perform and requires no extra equipment. Moreover, it can easily and safely measure large amounts of reactivity. On the other hand, the rod drop method is usually associated with reactor shutdown and subsequent reactor startup. Also, the rod drop time is not instantaneous as is theoretically assumed, therefore limiting the method’s accuracy. In PWRs, the drop time of all control rods is usually about 2 – 4 seconds.

This method is widely used to determine the worth of all control rods (i.e., of an emergency shutdown system). This method can also be used to determine the parameters of delayed neutrons, and these data can be obtained by decomposition of neutron flux coast-down.

A more accurate method is, for example, in:

Moore, K. V. Shutdown Reactivity by the Modified Rod Drop Method, USAEC Report ID0-16948, 1964.

Source Jerk Method
The source jerk method belongs to a group of source perturbation methods. But in principle, the source jerk method is essentially the same as the rod drop method, except that it is a subcritical measurement, and the neutron source is removed instead of inserting a control rod. This method is based on the study of the transient response of the reactor to a rapid neutron source removal from a reactor.

In this case, the subcriticality of the reactor (subcritical multiplication) can be determined from the measurement of the prompt drop. Also, here the prompt drop can be measured because the delayed neutron precursor population will not change immediately. Following sudden source removal, the neutron population undergoes a sharp negative jump.

As was described in the Prompt Jump Approximation, the response of a neutron detector (n1 ➝ n2) immediately after a source removal from a subcritical reactor (ρ1 < 0) is related by:

where n0 is the neutron level with the source in place and n1 is the neutron level immediately after the source jerk. Since a source is much smaller and lighter than a control rod, it is easier to remove it from the core quickly. On the other hand, in commercial reactors, this method cannot be used because, in commercial reactors, neutron sources (when used) cannot be removed from the core. Moreover, commercial reactors contain high burnup fuel, which is an important source of neutrons.

Reactivity

In the preceding chapters, the classification of states of a reactor according to the effective multiplication factor – keff was introduced. The effective multiplication factor – keff is a measure of the change in the fission neutron population from one neutron generation to the subsequent generation. But sometimes, it is convenient to define the change in the keff alone, the change in the state, from the criticality point of view.

For these purposes, reactor physics uses a term called reactivity rather than keff to describe the change in the state of the reactor core. The reactivity (ρ or ΔK/K) is defined in terms of keff by the following equation:

From this equation, it may be seen that ρ may be positive, zero, or negative. The reactivity describes the deviation of an effective multiplication factor from unity. For critical conditions, the reactivity is equal to zero. The larger the absolute value of reactivity in the reactor core, the further the reactor is from criticality. The reactivity may be used to measure a reactor’s relative departure from criticality.

It must be noted the reactivity can also be calculated according to another formula.

This formula is widely used in neutron diffusion or neutron transport codes. The advantage of this reactivity is obvious, it is a measure of a reactor’s relative departure not only from criticality (keff = 1), but it can be related to any sub or supercritical state (ln(k2 / k1)). Another important feature arises from the mathematical properties of the logarithm. The logarithm of the division of k2 and k1 is the difference between the logarithm of k2 and the logarithm of k1. ln(k2 / k1) = ln(k2) – ln(k1). This feature is important in the case of addition and subtraction of various reactivity changes.

See more: D.E.Cullen, Ch.J.Clouse, R.Procassini, R.C.Little. Static and Dynamic Criticality: Are They Different?. Lawrence Livermore National Laboratory. UCRL-TR-201506. 11/2003.

Units of Reactivity
Mathematically, reactivity is a dimensionless number, but it can be expressed by various units. The most common units for research reactors are units normalized to the delayed neutron fraction (e.g., cents and dollars) because they exactly express a departure from prompt criticality conditions.

The most common units for power reactors are units of pcm or %ΔK/K. The reason is simple. Units of dollars are difficult to use because the normalization factor, the effective delayed neutron fraction, significantly changes with the fuel burnup. In LWRs, the delayed neutron fraction decreases with fuel burnup (e.g., from βeff = 0.007 at the beginning of the cycle up to βeff = 0.005 at the end of the cycle). This is due to isotopic changes in the fuel. It is simple, and 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 for thermal fission), the resultant core delayed neutron fraction of a multiplying system decreases (the weighted average of the constituent delayed neutron fractions).

βcore= ∑ Pii

Inverse Reactor Kinetics – Reactimeter

The reactivity describes the measure of a reactor’s relative departure from criticality. During reactor operation and reactor startup, it is important to monitor the reactivity of the system. It must be noted reactivity is not directly measurable, and therefore most power reactors procedures do not refer to it, and most technical specifications do not limit it. Instead, they specify a limiting rate of neutron power rise (measured by excore detectors), commonly called a startup rate (especially in the case of PWRs).

On the other hand, during reload startup physics tests performed at the startup after refueling the commercial PWRs, it is important to monitor subcriticality continuously during the criticality approach. On-line reactivity measurements are based on the inverse kinetics method, and the inverse kinetics method is a reactivity measurement based on the point reactor kinetics equations. This method can be used for:

• Reactivity measurement at high neutron levelreactimeter without source term. A reactimeter can be constructed without source term, but it works only at higher neutron levels, where the neutron source term in point reactor kinetics equations may be neglected.
• Reactivity measurement at subcritical multiplicationreactimeter with source term. For operation at low power levels or in the sub-critical domain (e.g., during criticality approach), the contribution of the neutron source must be taken into account, and this implies the knowledge of a quantity proportional to the source strength and then it should be determined. The subcritical reactimeter is based on the determination of the source term (source strength).

As was written, the reactivity of the system can be measured by a reactimeter. The reactimeter is a device (or rather a computational algorithm) that can continuously give real-time reactivity using the inverse kinetics method. The reactimeter usually processes the signal from source range excore neutron detectors and calculates the reactivity of the system.

It was shown that the source term is not easy to determine, and the problem is that it is of the highest importance in the subcritical domain. One recognized method for source term determining is known as Least Squares Inverse Kinetics Method (LSIKM).

Special reference: Seiji TAMURA, “Signal Fluctuation and Neutron Source in Inverse Kinetics Method for Reactivity Measurement in the Sub-critical Domain,” J. Nucl. Sci. Technol, Vol.40, No. 3, p. 153–157 (March 2003)

Inverse Kinetics - Equations
The exact point kinetics equations that can be derived from the general neutron balance equations without making any approximations are:

where:

• n(t) is the neutron density in the core (which is proportional to the detector count rate)
• Ci(t) is the precursor’s density of delayed neutrons of group i
• Λ is the prompt generation time
• ρ is the reactivity
• the constants βi and λi are the fraction and decay constant of delayed neutron precursor of group i
• S(t) is the source term, which characterizes the number of neutrons (source neutrons) added to the system from an external source. As can be seen, the source term does not influence the system’s dynamics since it does not influence the reactivity of the system.

According to Seiji TAMURA, the Inverse Kinetics equations (flux ⟶ reactivity) for discrete-time series data can be derived from the ordinary point kinetics equations (reactivity ⟶ flux), assuming that the reactor power change for the interval of Δt is as n(t)=nj-1exp(μjt), where μj = log(nj/nj-1)/Δt:

When the reactor power is at a steady-state no, there are seven initial conditions that can be obtained as:

The equation for ρj provides a real-time reactivity calculation for successive reactor power data. When the reactor is operating at a sufficiently high power level, the last term of the right side, the source term, may be neglected because it becomes negligible.

For operation at low power levels or in the sub-critical domain (e.g., during criticality approach), the contribution of the neutron source must be taken into account, and this implies the knowledge of a quantity proportional to the source strength and then it should be determined. Otherwise, zero reactivity will be obtained for any subcritical reactor condition (after the transition phenomena have died out). The source term can be determined using data from a transient state after introducing a given reactivity. The source term can be determined from the rod drop experiment data using the Least Squares Inverse Kinetics Method (LSIKM).

Special reference: M. Itagaki, A. Kitano, “Revised source strength estimation for inverse neutron kinetics,” Preprints 1999 Fall Mtg., At. Energy Soc. Jpn., Hiroshima, Japan, G20, (1999)

Special reference: Renato Yoichi Ribeiro Kuramoto, Anselmo Ferreira Miranda. SUBCRITICAL REACTIVITY MEASUREMENTS AT ANGRA 1 NUCLEAR POWER PLANT. 2011 International Nuclear Atlantic Conference – INAC 2011. ISBN: 978-85-99141-04-5.

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