## Reactor Dynamics

In preceding chapters (Nuclear Chain Reaction), the classification of states of a reactor according to **the effective multiplication factor – k**** _{eff}** was introduced. The effective multiplication factor –

**k**

**is a measure of the change in the fission**

_{eff}**neutron population**from one neutron generation to the subsequent generation. Also the reactivity as a measure of a reactor’s relative departure from criticality was defined.

In this section, amongst other things it will be briefly described how **the neutron flux** (i.e. the reactor power) changes if **reactivity** of a multiplying system is not equal to zero. We will study the **time-dependent behaviour** of nuclear reactors. An understanding of 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 of the most importance in the nuclear reactor safety. This chapter is named the **Reactor Dynamics**, but also comprises the **reactor kinetics**. **Nuclear reactor kinetics** is dealing 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 due to 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 feedbacks**.**Reactor Dynamics.**Reactor dynamics is the study of 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****feedbacks**and with spatial effects.

Time-dependent behaviour of nuclear reactors can be also 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 over the course of 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 as a result of**fuel burnup**.

This chapter is concerned with short-, medium- and long-term kinetics, despite the fact the fuel burnup and other changes in fuel composition are usually not a dynamic problem. At first, we have to start with an introduction to **prompt and delayed neutrons** because they play an important role in short-term reactor kinetics. 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.

## Prompt and Delayed Neutrons

It is known the fission neutrons are of importance in any chain-reacting system. Neutrons trigger the nuclear fission of some nuclei (^{235}U, ^{238}U or even ^{232}Th). What is crucial the fission of such nuclei produces **2, 3 or more** free neutrons.

But not all neutrons are released **at the same time following fission**. Even the nature of creation of these neutrons is different. From this point of view we usually divide the fission neutrons into two following groups:

**Prompt Neutrons.**Prompt neutrons are emitted**directly from fission**and they are emitted within**very short time of about 10**.^{-14}second**Delayed Neutrons.**Delayed neutrons are emitted by**neutron rich fission fragments**that are called**the delayed neutron precursors**. These precursors usually undergo beta decay but a small fraction of them are excited enough to undergo**neutron emission**. The fact the neutron is produced via this type of decay and this happens**orders of magnitude later**compared to the emission of the prompt neutrons, plays an extremely important role in the control of the reactor.

## Key Characteristics of Prompt Neutrons

- Prompt neutrons are emitted
**directly from fission**and they are emitted within very short time of about**10**.^{-14}second

- Most of the neutrons produced in fission are prompt neutrons –
**about 99.9%**.

- For example a fission of
^{235}U by thermal neutron yields**2.43 neutrons**, of which 2.42 neutrons are prompt neutrons and 0.01585 neutrons are the delayed neutrons.

- The production of prompt neutrons slightly increase with incident neutron energy.

- Almost all prompt fission neutrons have
**energies between 0.1 MeV and 10 MeV**.

- The mean neutron energy is about
**2 MeV**. The most probable neutron energy is about**0.7 MeV**.

- In reactor design
**the prompt neutron lifetime**(PNL) belongs to key neutron-physical characteristics of reactor core.

- Its value depends especially on the type of the moderator and on the energy of the neutrons causing fission.

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

- In LWRs the
**PNL increases with the fuel burnup**.

- The typical prompt neutron lifetime in thermal reactors is on the order of
**10**second.^{-4}

- The typical prompt neutron lifetime in fast reactors is on the order of
**10**second.^{-7}

## Key Characteristics of Delayed Neutrons

- The presence of delayed neutrons is perhaps
**most important aspect of the fission process**from the viewpoint of reactor control.

- Delayed neutrons are emitted by neutron rich fission fragments that are called the
**delayed neutron precursors**.

- These precursors usually undergo beta decay but a small fraction of them are excited enough
**to undergo neutron emission.**

- The emission of neutron happens orders
**of magnitude later**compared to the emission of the prompt neutrons.

- About
**240 n-emitters**are known between^{8}He and^{210}Tl, about 75 of them are in the non-fission region.

- In order to simplify reactor kinetic calculations it is suggested
**to group together the precursors**based on their half-lives.

- Therefore delayed neutrons are traditionally represented by
**six delayed neutron groups**.

- Neutrons can be produced also in
**(γ, n) reactions**(especially in reactors with heavy water moderator) and therefore they are usually referred to as**photoneutrons**.**Photoneutrons**are usually treated no differently than regular delayed neutrons in the kinetic calculations.

- The total yield of delayed neutrons per fission, v
_{d}, depends on:- Isotope, that is fissioned.
- Energy of a neutron that induces fission.

- Variation among individual group yields is much greater than variation among group periods.

- In reactor kinetic calculations it is convenient to use relative units usually referred to as
**delayed neutron fraction (DNF)**.

- At the steady state condition of criticality, with k
_{eff}= 1, the delayed neutron fraction is equal to the precursor yield fraction β.

- In LWRs the
**β decreases with fuel burnup**. This is due to isotopic changes in the fuel.

- Delayed neutrons have
**initial energy between 0.3 and 0.9 MeV**with an**average energy of 0.4 MeV**.

- Depending on the
**type of the reactor**, and their**spectrum**, the delayed neutrons may be more (in thermal reactors) or less effective than prompt neutrons (in fast reactors). In order to include this effect into the reactor kinetic calculations**the effective delayed neutron fraction – β**must be defined._{eff}

- The effective delayed neutron fraction is the product of the average delayed neutron fraction and the importance factor
**β**_{eff}= β . I.

- The weighted delayed generation time is given by
**τ = ∑**, therefore the weighted decay constant_{i}τ_{i}. β_{i}/ β = 13.05 s**λ = 1 / τ ≈ 0.08 s**.^{-1}

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

- 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.

**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 a 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 **β _{eff}**(i.e.

**k**. In power reactors such a reactivity insertion is

_{eff}≈ 1.006 or ρ = +600 pcm)**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 a

**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.

## Point Kinetics Equations

As we have seen in previous chapters, the number of neutrons is multiplied by a factor k_{eff} from one neutron generation to the next, therefore the multiplication environment (nuclear reactor) behaves like the exponential system, that means the power increase is not linear, but it is **exponential**.

It is obvious** the effective multiplication factor** in a multiplying system is a measure of the change in the fission neutron population from one neutron generation to the subsequent generation.

**k**_{eff}**< 1**. If the multiplication factor for a multiplying system is**less than 1.0**, then the**number of neutrons is decreasing**in time (with the mean generation time) and the chain reaction will never be self-sustaining. This condition is known as**the subcritical state**.

**k**_{eff}**= 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**.

**k**_{eff}**> 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 is exponentially increasing 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**, that means,** how many times in a one second we have to multiply the neutron population by a factor k _{eff}**. 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 behaviour of the system. Therefore such a discussion will be not trivial.

To study the kinetic behaviour of the system, engineers usually use the **point kinetics equations**. The name **point kinetics** is used because, in this simplified formalism, the **shape** of the neutron flux and the neutron density **distribution** are **ignored**. The reactor is therefore **reduced to a point**. In the following section we will introduce the point kinetics and we start 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 to

**its escape**from the system. The average number of neutrons that disappear during a unit time interval

*dt*is

**But each disappearance of a neutron contributes an average of**

*n.dt/l.**k*new neutrons.

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

**where:**

**n(t) = transient reactor power**

**n(0) = initial reactor power**

**τ**_{e}** = reactor period**

**The reactor period, ****τ**** _{e}**, 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

**τ**

**, the**

_{e}**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 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. Shorter lifetimes give simply faster responses of 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 without delayed neutrons).

This simple point kinetics equation is often expressed is 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/k**_{eff}**= prompt neutron generation time,**which is the average time from a prompt neutron emission to an 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. During this transients the prompt neutron lifetime is not constant 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 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 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 e^{5} = 2.718^{5}, in 10 seconds the reactor would pass through 50 periods and the power would increase by e^{50} = ……

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

**T = l / (k**_{∞ }**– 1) = 10-7 / (1.0001 – 1) = 0.001s**

**Reactors with such a kinetics would be very difficult to control.** **Fortunately this behaviour 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 absorbtion**(fission or radiative capture) or to

**its escape**from the system. This parameter is defined in multiplying or also in nonmultiplying systems. In both systems the prompt neutron lifetimes depend strongly on:

- material composition of the system
- multiplying – nonmultiplying system
- system with or without thermalization
- isotopic composition of the system

- geometric configuration of the system
- homogeneous or heterogeneous system
- shape of entire system

- size of the system

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

**l=t _{s} + t_{d}**

In an infinite thermal reactor **t _{s} << t_{d}** and therefore

**l ≈ t**. The typical prompt neutron lifetime

_{d}**in thermal reactors**is on the order of

**10**. Generally, the longer neutron lifetimes take place in systems in which the neutrons must be thermalized in order to be absorbed.

^{−4}secondSystems in which most of the neutrons are absorbed at 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} second**.

In multiplying systems, in which 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 k_{eff}=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/k _{eff}**

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

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

**key neutron-physical characteristics**of reactor core. Its value depends especially on the type of the moderator and on the energy of the neutrons causing fission. Its importance for nuclear reactor safety is well known for a long time.

The longer prompt neutron lifetimes can substantially improve kinetic response of reactor (**the longer prompt neutron lifetime gives simply 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 or can be modified in order to increase the PNL and in order to 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**** (l**_{d}**)**.

**l**_{d}** = (1 – β).l**_{p}** + ∑l**_{i}** . β**_{i}** => l**_{d}** = (1 – β).l**_{p}** + ∑τ**_{i}** . β**_{i}

where

**(1 – β)**is the fraction of all neutrons emitted as prompt neutrons**l**is the prompt neutron lifetime_{p}**τ**is the mean precursor lifetime, the inverse value of the decay constant_{i }**τ**_{i}**= 1/λ**_{i}- The weighted delayed generation time is given by
**τ = ∑τ**_{i}**. β**_{i}**/ 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 have

**a dominating effect of 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:

**l**_{d}** = (1 – β).l**_{p}** + ∑τ**_{i}** . β**_{i}** = (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 **l**** _{d}** is evident. Longer lifetimes give simply slower responses of multiplying systems. The role of reactivity (k

_{eff}– 1) is also evident. Higher reactivity gives simply larger response of 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 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 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 = l**_{d}** / (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 completely different dimension of the response on reactivity insertion in comparison with the case without presence of delayed neutrons, where the reactor period was 1 second.

Reactor-kinetic calculations with considering of such a number of initial conditions would be correct, but it also would 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 to experimental measurements. This model has 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.

- These constants are different for each fissionable nuclide.

- These constants change also with the neutron energy spectrum.

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

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

**reactor kinetics**and in

**a subcriticality control**. Especially in nuclear reactors with D

_{2}O moderator (CANDU reactors) or with Be reflectors (some experimental reactors). Neutrons can be produced also in

**(γ, n) reactions**and therefore they are usually referred to as

**photoneutrons**.

A high energy photon (gamma ray) can under certain conditions **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 in excess of **6 MeV**, which is above the energy of most gamma rays from fission. On the other hand **there are few nuclei** with sufficiently low binding energy to be of **practical interest**. These are: ** ^{2}D, ^{9}Be**,

^{6}Li,

^{7}Li and

^{13}C. As can be seen from the table

**the lowest threshold**have

**and**

^{9}Be with 1.666 MeV**.**

^{2}D with 2.226 MeVIn 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 regular delayed neutrons in the kinetic calculations. Photoneutron precursors can be also 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.

Despite the fact photoneutrons are of importance especially in CANDU reactors, deuterium nuclei are always present **(~0.0156%**) **also 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 D_{2}O. This enhances the heavy water concentration. Therefore also in LWRs kinetic calculations, photoneutrons from D_{2}O are treated as additional groups of delayed neutrons having characteristic decay constants λ_{j} and effective group fractions.

After a nuclear reactor has been operated 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**. **In the long term** the photoneutron source decreases with the** decay of long-lived fission products** that produce delayed high-energy gamma rays and the photoneutron source drops slowly, decreasing a little each day. The longest-lived fission product with gamma ray energy above the threshold is ** ^{140}Ba** with a half-life of

**12.75 days.**

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

In comparison with **fission neutrons**, that make a **self-sustaining chain reaction possible**, **delayed neutrons** make reactor **control possible** and **photoneutrons** are of importance **at low power operation**.

See also: Effective Delayed Neutron Fraction – βeff

The delayed neutron fraction, **β**, is the fraction of delayed neutrons in the core **at creation, that is, at high energies**. But in case of thermal reactors the fission can be initiated **mainly by thermal neutron**. Thermal neutrons are of practical interest in study of thermal reactor behaviour. **The effective delayed neutron fraction**, usually referred to as **β _{eff}**, 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 **β _{eff}** 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 in thermal reactors a delayed neutron **traverses a smaller energy range** to become thermal and it is also **less likely to be lost** by leakage or by parasitic absorption than is 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 forms 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.

**β _{eff} = 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 differs also from group to group, the different groups of delayed neutrons will also have a different effectiveness. Moreover, a nuclear reactor contains a mixture of fissionable isotopes. Therefore, in some cases, the importance factor is insufficient 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 **β _{eff}** is less than β by about 10%.

**Mean Generation Time with Delayed Neutrons**,

**l**, is the weighted average of the prompt generation times and a delayed neutron generation time. The delayed neutron generation time,

_{d}**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 down time and the diffusion time) is very short compared with the mean lifetime of their precursors (t_{s} + t_{d} << τ_{i}). Therefore τ_{i} is also equal to the mean lifetime of a neutron from the ith group, that is, **τ _{i} = l_{i}** and the equation for mean generation time with delayed neutrons is the following:

**l _{d} = (1 – β).l_{p} + ∑l_{i} . β_{i} => l_{d} = (1 – β).l_{p} + ∑τ_{i} . β_{i}**

where

**(1 – β)**is the fraction of all neutrons emitted as prompt neutrons**l**is the prompt neutron lifetime_{p}**τ**is the mean precursor lifetime, the inverse value of the decay constant_{i }**τ**_{i}= 1/λ_{i}- The weighted delayed generation time is given by
**τ = ∑τ**_{i}. β_{i}/ 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 have

**a dominating effect of 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:

**l _{d} = (1 – β).l_{p} + ∑τ_{i} . β_{i} = (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.

**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 a 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 **β _{eff}**(i.e.

**k**. In power reactors such a reactivity insertion is

_{eff}≈ 1.006 or ρ = +600 pcm)**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 a

**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.

**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 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,

**. 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**

*-n/l***production and the delayed channel resulting from radioactive decay of precursor nuclei (λ**

*(1-β)*_{i}C

_{i}).

In the **equation for precursors**, there is the balance between the production of the precursors of i-th group and their decay after the decay constant λ_{i}. As can be seen, the rate of the decay of precursors is the radioactivity rate (λ_{i}C_{i}) and the rate of production is proportional to the number of neutrons times **β**_{i}**, which **is defined as the fraction of the neutrons which appear as **delayed neutrons in the i th 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 is 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. During this transients the prompt neutron lifetime is not constant 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**. In fact the reactivity may be used as a measure of a **reactor’s relative departure from criticality**. According to the reactivity, we can classify the different reactor states and the related consequences as follows:

**The prompt critical state is defined as:**

**k**_{eff}** > 1; ρ ≥ β**** _{eff}**, 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**

^{-5}**s**).

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

**k _{eff} > 1; 0 < ρ < β_{eff}**, 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 in order to sustain the chain reaction. The neutron population increases, but

**much more slowly**(as the mean generation lifetime with delayed neutrons ~0.1 s).

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

**k _{eff} = 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 in order 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 state as the critical state from basic classification.

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

**k _{eff} < 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 **inverse hour**, when it was 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 external source in which the properties are changed at t = 0 in such a way as to introduce a **step reactivity ρ _{0}** 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

**C**(equations for the density of precursors).

_{i}(t) = C_{i,0}.exp(s.t)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** s _{0 – 6}** was originally determined in inverse hours. For a given value of the reactivity

**ρ**the associated values of

**s**are determined with this equation. The following figure shows the relation between

_{0 – 6}**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 s**. The remaining terms rapidly die away, yielding an asymptotic solution in the form:

_{0}is positivewhere **s _{0} = 1/τ_{e}** is the

**stable reactor period**or

**asymptotic period of reactor**. This root,

**s**, is

_{0}**positive for ρ > 0**and

**negative for ρ < 0**, therefore this root describes the reactor response, which is lasting after the transition phenomena have died out. The figure also shows that a negative reactivity leads to a negative period: All of the s

_{i}are negative, but the root s

_{0}will die away more slowly than the others. Thus the solution

**n(t) = A**is valid for positive as well as negative reactivity insertions.

_{0}exp(s_{0}t)To determine the reactivity required to produce a given period a plot of ρ vs. τ_{e} must be constructed using the delayed neutron data for a particular fissionable isotope or mix of isotopes, and for a given prompt generation time. To determine the stable reactor period, which results from a given reactivity insertion, it is convenient to use the following form of inhour equation.

where:

**β**** _{eff}** = effective delayed neutron fraction

**λ**** _{eff}** = effective delayed neutron precursor decay constant

**τ**** _{e}** = reactor period

**ρ** = reactivity

The first term in this formula is the **prompt term** and it causes that the positive reactivity insertion is followed immediately by a immediate power increase called the **prompt jump**. This power increase occurs because the rate of production 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.

**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 case of PWRs).

**The reactor startup rate** is defined as the number of factors of ten 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 following equation:

**n(t) = n(0).10 ^{SUR.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 is increasing. If SUR is negative, reactor power is decreasing. The relationship between reactor period and startup rate is given by following equations:

Example:

Suppose **k _{eff} = 1.0005** in a reactor with a generation time

**l**. For this state calculate the reactor period –

_{d}= 0.01s**τ**, doubling time –

_{e}**DT**and the startup rate (

**SUR**).

ρ = 1.0005 – 1 / 1.0005 =

**50 pcm**

τ_{e} = l_{d} / k-1 = 0.1 / 0.0005 = **200 s**

DT = τ_{e} . ln2 = **139 s**

SUR = 26.06 / 200 = **0.13 dpm**

## Special Cases of Inhour Equation

**ρ is very small.**This causes the first root, s

_{0}, of the reactivity to be also very small. In this case,

**s**

_{0}**≪**

**λ**

**and therefore, we can ignore the term**

_{i}**s**

**in the denominator of the inhour equation. The**

_{0}**inhour equation**then takes form:

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

Thus for small reactivities—positive or negative—the reactor period is governed almost completely by the delayed neutron properties. Despite the fact the **amount of delayed neutrons** is only on the order of **tenths of percent** of the total amount, **the timescale in seconds (**τ_{i}**) plays the extremely important role.**

The assumption is **s**_{0}**≪** **λ**_{i}**. **The largest value of **τ**_{i}** = 1/λ**_{i}** = 80s, **therefore the this formula is valid for periods, τ_{e}, 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:

**Mean generation time with delayed neutrons**** (l**_{d}**)**:

**l**_{d}** = (1 – β).l**_{p}** + ∑l**_{i}** . β**_{i}** => l**_{d}** = (1 – β).l**_{p}** + ∑τ**_{i}** . β**_{i}

where

**(1 – β)**is the fraction of all neutrons emitted as prompt neutrons**l**is the prompt neutron lifetime_{p}**τ**is the mean precursor lifetime, the inverse value of the decay constant_{i }**τ**_{i}**= 1/λ**_{i}- The weighted delayed generation time is given by
**τ = ∑τ**_{i}**. β**_{i}**/ 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 have

**a dominating effect of 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:

**l**_{d}** = (1 – β).l**_{p}** + ∑τ**_{i}** . β**_{i}** = (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 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 = l**_{d}** / (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 completely different dimension of the response on reactivity insertion in comparison with the case without presence of delayed neutrons, where the reactor period was 1 second.

**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**

^{-5}**s**).

This causes the first root, **s _{0}**, of the reactivity becomes large compared to each

**λ**

**. In this case,**

_{i}**λ**

_{i}**≪**

**s**

**and therefore, we can ignore the term**

_{0}**λ**

**in the denominator of the**

_{i}**inhour equation**. The inhour equation then takes 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**.

Obviously, 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, for now the prompt neutron lifetime rather than the delayed neutron half-lives largely determines the rate of exponential increase. Reactors with such a 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 a 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 kinetic response of reactor (**the longer prompt neutron lifetime gives simply slower power increase**). Therefore the PNL should be verified in a reload safety evaluation (RSE) process. Management of this accident is based on reactivity feedbacks, especially on the **doppler temperature coefficient (DTC)**. This coefficient is of the highest importance in the **reactor stability**. The **doppler temperature coefficient** is generally considered to be even **more important** than the **moderator temperature coefficient**** (MTC)**. Especially in case of a control rod ejection the doppler temperature coefficient will be **the first** and the most important feedback, that will compensate 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** on changes in fuel temperature.

**ρ is negative and ρ**

**≪**

**β**

**.**This case is typical for

**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:

It is obvious, the neutron flux cannot drop below the value *βn*_{1}**. ** 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 = -15 β = -0.09 (β= 600pcm = 0.006)**

For this negative reactivity the prompt drop is equal to:

**n _{2}/n_{1}** = 0.006/(0.006+0.09)=

**0.063**

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

The neutron flux then continues to fall according to stable period. The first root of reactivity equation occurs at **s**_{0}** = – λ**_{1}**. **the decay constant of the long-lived precursors group. The shortest negative stable period is then **τ _{e} = – 1/λ**

_{1}**= -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 and within few tens of seconds the thermal power which originates from nuclear fission is below the thermal power which originates from decay heat.

## Reactivity Pulse – Impulse Characteristics

We will now study a response of a reactor on a **reactivity pulse**, which is represented by the **Dirac delta function**, δ(t). Strictly speaking 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) = ρ_{0} . 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 following relations:

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

## Oscillation of Reactivity – Frequency Characteristics

We will now study a response of a reactor on a **reactivity oscillation**, which is represented by the following function: ** ρ(t) = ρ_{0} . cos(ωt)**. Where

*ρ*is the amplitude of the input signal (forcing function) and ω is the signal frequency expressed in radians per second.

_{0}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 behaviour of a reactor. The exact solution can be obtained relatively easy 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

**ρ**<

**) 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 point kinetics equation.**

*β*If we are interested in **long-term behaviour** (asymptotic period) and not interested in the details of the prompt jump, 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)** in which the rapid power change due to prompt neutrons 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. C_{i,1} = C_{i,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 < 0.5 β. **It is usually used with another simplification, the

**one delayed precursor group approximation**.

**prompt jump approximation (PJA)**. This eliminated the fast time scale due to prompt neutrons. In this section we consider 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:

This simplification then leads to:

Assuming that the reactivity is constant and n_{1}/n_{0} can be determined from prompt jump formula this equation leads to very simple formula:

**ρ**<

**) 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 point kinetics equation.**

*β*If we are interested in short-term behaviour and not interested in the details of the asymptotic behaviour, 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 amount of delayed neutrons are neglected, corresponding to taking dC_{i}(t)/dt = 0 and C_{i}(t) = C_{i,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. On the other hand they cannot be used to determine most dynamic characteristics.**Kinetic methods**are used to study parameters (parameters of delayed neutrons etc.) which determine short-term and medium-term kinetics.

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

**stable reactor period (or asymptotic period),**

**τ**

**, is defined as the time required for the neutron density to change by a factor e = 2.718.**

_{e}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 C_{i}(t) = C_{i,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 s_{0} = 1/_{e} is the stable reactor period or asymptotic period of reactor. This root, s_{0}, is positive for ρ > 0 and negative for ρ < 0, therefore this root describes the reactor response, which is lasting after the transition phenomena have died out. The measurement of asymptotic period, can be used to determine (directly from inhour equation) the reactivity inserted to the system.

**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 the dropping of 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 (n_{1} ➝ n_{2}) immediately after a control rod is dropped into a critical reactor (ρ_{1} = 0) is related by:

which allows determination of the reactivity worth of the rod. The **rod drop method** is advantageous because it is very quick to perform and it requires no extra equipment. Moreover it can easily and safely measure large amounts of reactivity. On the other hand the rod drop method is associated usually with reactor shutdown and subsequent reactor startup. Also, the rod drop time is not instantaneous as is theoretically assumed, therefore limiting the accuracy of the method. 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 be also used to determine the parameters of delayed neutrons. These data can be obtained by decomposition of neutron flux coastdown.

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**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 (n_{1} ➝ n_{2}) immediately after a source removal from a subcritical reactor (ρ_{1} < 0) is related by:

where n_{0} is the neutron level with the source in place and n_{1} is the neutron level immediately after the source jerk. Since a source is much smaller and lighter than a control rod, it is much easier to quickly remove from the core. On the other hand, in commercial reactors, this method cannot be used, because in commercial reactors neutron sources (when used) cannot be simply removed from the core. Moreover commercial reactors contains high burnup fuel which is very important source of neutrons.

See also: Source Neutrons

## Reactivity

In preceding chapters, the classification of states of a reactoraccording to the effective multiplication factor – k_{eff} was introduced. The effective multiplication factor – k_{eff} 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 k _{eff}** alone, the change in the state, from the criticality point of view.

For these purposes reactor physics use a term called **reactivity** rather than k_{eff} to describe the change in the state of the reactor core. **The reactivity** (**ρ** or** ΔK/K**) is defined in terms of k_{eff} 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**. In fact the reactivity may be used as a measure of a **reactor’s relative departure from criticality**.

It must be noted the reactivity can be calculated also according to the 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 (k_{eff} = 1), but it can be related to any sub or supercritical state (**ln(k _{2} / k_{1})**). Another important feature arises from the

**mathematical properties of logarithm**. The logarithm of the division of k

_{2}and k

_{1}is the difference of logarithm of k

_{2}and logarithm of k

_{1}.

**ln(k**. This feature is important in case of addition and subtraction of various reactivity changes.

_{2}/ k_{1}) = ln(k_{2}) – ln(k_{1})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.

**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 **β _{eff} = 0.007** at the beginning of the cycle up to

**β**at the end of the cycle). This is due to isotopic changes in the fuel. It is simple,

_{eff}= 0.005**fresh uranium fuel**contains only

^{235}U as the fissile material, meanwhile during fuel burnup the importance of fission of

^{239}Pu increases (in some cases up to 50%). Since

^{239}Pu produces significantly less delayed neutrons (

**0.0021**for thermal fission), the resultant core delayed neutron fraction of a multiplying system decreases (it is the weighted average of the constituent delayed neutron fractions).

β_{core}= ∑ P_{i}.β_{i}

## Inverse Reactor Kinetics – Reactimeter

The reactivity describes the measure of a **reactor’s relative departure from criticality**. During reactor operation and during 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 case of PWRs).

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

**Reactivity measurement at high neutron level**–**reactimeter without source term**. Without source term a reactimeter can be constructed, 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 multiplication**–**reactimeter 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 so easy to be determined and the problem is that the source term is of the highest importance in the subcritical domain. One of the recognize methods 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)

**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)
- C
_{i}(t) is the precursors density of delayed neutrons of group i - Λ is the prompt generation time
- ρ is the reactivity
- the constants β
_{i}and λ_{i}are the fraction and decay constant of delayed neutron precursor of group i - S(t) is the source term, which characterizes number of neutrons (source neutrons) added to the system from an external source. As can be seen, the source term does not influence the dynamics of the system, 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)=n_{j-1}exp(μ_{j}t), where μ_{j} = log(n_{j}/n_{j-1})/Δt:

When the reactor power is at a steady state n_{o}, there are seven initial conditions that can be obtained as:

The equation for ρ_{j} provides a real time reactivity calculation for successive reactor power data. When the reactor is operating at 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 the introduction of a given reactivity. The source term can be determined from data from rod drop experiment 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.

## Reactivity Coefficients – Reactivity Feedbacks

Up to this point, we have discussed the response of the **neutron**

** population** in a **nuclear reactor** to an **external reactivity input**. There was applied an assumption that the level of the neutron population **does not affect** the properties of the system, especially that the neutron power (power generated by chain reaction) is sufficiently **low** that the reactor core does not change its **temperature** (i.e. **reactivity feedbacks may be neglected**). For this reason such treatments are frequently referred to as the **zero-power kinetics**.

However, in an operating **power reactor** the neutron population is always large enough to generated heat. In fact, it is the main purpose of power reactors **to generate large amount of heat**. This causes the temperature of the system changes and material densities change as well (due to the **thermal expansion**).

Demonstration of the** prompt negative temperature coefficient** at the **TRIGA reactor**. A major factor in the prompt negative temperature coefficient for the TRIGA cores is the core spectrum hardening that occurs as the fuel temperature increases. This factor allows TRIGA reactors to operate **safely** during either **steady-state** or **transient conditions**.

Source: Youtube

See also: General Atomics – TRIGA

Because macroscopic cross sections are proportional to densities and temperatures, **neutron flux spectrum** depends also on the density of moderator, these changes in turn will produce some changes in reactivity. These changes in reactivity are usually called the **reactivity feedbacks** and are characterized by **reactivity coefficients**. This is very important area of reactor design, because the reactivity feedbacks influence the** stability of the reactor**. For example, reactor design must assure that under all operating conditions the temperature feedback will be **negative**.

## How negative feedback acts against power excursion

### Example: Change in the moderator temperature.

**Negative feedback** as the moderator temperature effect influences the neutron population in the following way. If the temperature of the moderator is increased, negative reactivity is added to the core. This negative reactivity causes reactor power to decrease. As the thermal power decreases, the power coefficient acts against this decrease and the reactor returns to the critical condition. The reactor power stabilize itself. In terms of multiplication factor this effect is caused by significant changes in the resonance escape probability and in the total neutron leakage (or in the thermal utilisation factor when chemical shim is used).

**↑T**

_{M}⇒ ↓k_{eff}= η.ε. ↓p . ↑f . ↓P_{f }. ↓P_{t }(BOC)**↑T _{M} ⇒ ↓k_{eff} = η.ε. ↓p .f. ↓P_{f }. ↓P_{t } (EOC)**

**Resonance escape probability.** It is known, the resonance escape probability is dependent also on the **moderator-to-fuel ratio**. As the moderator temperature increases the ratio of the moderating atoms (molecules of water) decreases as a result of the **thermal expansion** of water. Its density decreases. This, in turn, causes a **hardening of neutron spectrum** in the reactor core resulting in higher resonance absorption (lower p). Decreasing the density of the moderator causes that **neutrons stay at a higher energy for a longer period**, which increases the probability of non-fission capture of these neutrons. This process is one of two processes (or three if chemical shim is used), which determine the moderator temperature coefficient.

**↑T**

_{M}⇒ ↓k_{eff}= η.ε. ↓p . ↑f . ↓P_{f }. ↓P_{t }(BOC)**↑T _{M} ⇒ ↓k_{eff} = η.ε. ↓p .f. ↓P_{f }. ↓P_{t } (EOC)**

**Thermal utilization factor. **The impact on the thermal utilization factor depends strongly on the amount of boron which is diluted in the primary coolent (chemical shim).** **As the moderator temperature increases the density of water decreases due to the **thermal expansion** of water. But along with the moderator also **boric acid is expanded** out of the core. Since boric acid is a neutron poison, and it is expanding out of the core, positive reactivity is added. The positive reactivity addition due to the expansion of boron out of the core offsets the negative reactivity addition due to the expansion of the moderator out of the core. It is obvious this effect is significant **at the beginning of the cycle** (BOC) and gradually loses its significance as the boron concentration decreases.

**↑T**

_{M}⇒ ↓k_{eff}= η.ε. ↓p . ↑f . ↓P_{f }. ↓P_{t }(BOC)**↑T _{M} ⇒ ↓k_{eff} = η.ε. ↓p .f. ↓P_{f }. ↓P_{t } (EOC)**

**Change of the neutron leakage. **Since both (**P _{f} and P_{t}**) are affected by a change in

**moderator temperature**in a heterogeneous water-moderated reactor and the directions of the feedbacks for both negative, the resulting

**total non-leakage probability**is also sensitive on the change in the moderator temperature. In result, an

**increase in the moderator temperature**causes that the probability of

**leakage increases**. In case of

**the fast neutron leakage,**the moderator temperature influences macroscopic cross-sections for elastic scattering reaction (Σ

_{s}=σ

_{s}.N

_{H2O}) due to the thermal expansion of water, which results in an increase in the moderation length. This, in turn, causes an increase of the leakage of fast neutrons.

- For
**the thermal neutron leakage**there are two effects. Both processes have the same direction and together causes the increase in the thermal neutron leakage. This physical process is a part of the**moderator temperature coefficient (MTC).**- Macroscopic cross-sections for elastic scattering reaction
**Σ**which significantly changes due to the_{s}=σ_{s}.N_{H2O,}**thermal expansion**of water. As the temperature of the core increases,**the diffusion coefficient**(**D = 1/3.Σ**) increases._{tr} - Microscopic cross-section (
**σ**) for neutron absorption changes with core temperature. As the temperature of the core increases, the absorption cross-section decreases._{a}

- Macroscopic cross-sections for elastic scattering reaction

This figure shows the power excursion as a result of positive reactivity on a logarithmic scale. There is a curve without feedback, along with a curve for the same reactivity insertion but for which the effects of negative temperature feedback are included. It can be seen both curves initially follow the same, but as the power becomes larger the curve with feedback becomes concave downward and stabilizes at a constant power. At this point the negative feedback has completely compensated for the initial reactivity insertion.

### Examples: Change in the reactor power

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 boron dilution). 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%**

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 **100%** of rated power and that the plant operator wants to decrease power to **75%** of rated power. The reactor operator must first bring the reactor subcritical by insertion of a negative reactivity (e.g. by control rod insertion or boric acid addition). As the thermal power decreases, moderator temperature and fuel temperature decrease as well, causing a positive reactivity effect (from the power coefficient) and the reactor returns to the critical condition. In order to keep the power to be decreasing, **negative 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.

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**

**100% → ↓ 20 steps or ↑ 18 ppm of boric acid within 10 minutes → 90%→ next ↓ 20 steps or ↑ 18 ppm within 10 minutes → 80% → final ↓ 10 steps or ↑ 9 ppm within 5 minutes→ 75%**

In order to describe the influence of all these processes on the reactivity, one defines the **reactivity coefficient α**. A reactivity coefficient is defined as the change of reactivity per unit change in some operating parameter of the reactor. For example:

α = ^{dρ}⁄_{dT}

The amount of reactivity, which is inserted to a reactor core by a specific change in an operating parameter, is usually known as the **reactivity effect** and is defined as:

dρ = α . dT

**The reactivity coefficients** that are important in power reactors (PWRs) are:

**Moderator Temperature Coefficient – MTC****Fuel Temperature Coefficient or Doppler Coefficient****Pressure Coefficient****Void Coefficient**

As can be seen, there are not only **temperature coefficients** that are defined in reactor dynamics. In addition to these coefficients, there are two other coefficients:

The total power coefficient is the combination of various effects and is commonly used when reactors are at power conditions. It is due to the fact, at power conditions it is difficult to separate the moderator effect from the fuel effect and the void effect as well. All these coefficients will be described in following separate sections. The reactivity coefficients are of importance in safety of each nuclear power plant which is declared in the **Safety Analysis Report** (SAR).

## Feedback Delay – Time Constants

In physics and engineering, the **time constant **is the parameter characterizing the response to a step input of a first-order, linear time-invariant system. The time constant is usually denoted by the Greek letter τ (tau). It is obvious, that thermal time constants, which characterize time required to warm an object by another object, are of importance for **reactor stability**. In general, the heat transfer from the body to the ambient at a given time is proportional to the temperature difference between the body and the ambient. The time constants that determine the time delays for reactivity feedbacks depend on the specific reactor design. For LWRs the following time constants are usual:

- The
**time constant**for**heating fuel is almost zero**, therefore the fuel temperature coefficient is effective almost instantaneously. - The
**time constant**for**heat transfer out of a fuel pin**varies from a few tenths to a few tens of seconds. The time for heat to be transferred to the moderator is usually measured in seconds (~5s). The presence of the surface film increases the time constant for the fuel element. - The time constant for equalization of temperatures within the primary loop depends strongly on length of primary piping and the flow velocity. It is usually measured in seconds (<10s)

Knowledge of delays of **reactivity feedbacks** is very important in the study of **reactor transients**. A large reactivity insertion will cause a reactor to go on such a short period that the power changes significantly over time spans that are short compared to the number of seconds needed to transfer heat from fuel to coolant. Especially in case of all **reactivity initiated accidents **(RIA), over short time spans the amount of heat transferred to the coolant is not enough to increase its temperature appreciably. Therefore, the fuel temperature coefficient will be **the first** and the most important feedback, that will compensate 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** on changes in fuel temperature.

## Point Dynamics Equations

A simple **point dynamics model** is based on point kinetics equations, but here we should take into account the influence of the fuel and the moderator temperature on the reactivity. We assume, there are no other feedbacks and therefore this simple model can be applied only on PWRs. For example, the void coefficient is here neglected. In systems with **boiling conditions**, such as boiling water reactors (BWR), the void coefficient is of prime importance during reactor operation.

Thus, the simplest point dynamics model of PWR should take into consideration the time variations of the fuel and coolant temperature. The point dynamics model consist of the following equations:

- The first equation is the
**equation for neutrons.**The first term on the right hand side is the production of prompt neutrons in the present generation, minus the total number of neutrons in the preceding generation. The second term is the production of delayed neutrons in the present generation. - The second equation is the
**equation for precursors**. There is the balance between the production of the precursors of i-th group and their decay after the decay constant λ_{i}. As can be seen, the rate of the decay of precursors is the radioactivity rate (λ_{i}C_{i}) and the rate of production is proportional to the number of neutrons times**β**_{i}**, which**is defined as the fraction of the neutrons which appear as**delayed neutrons in the i**.*th*group - The third equation expresses the dependence of the reactivity on various parameters. But in this case, there is a dependence on the coolant and the fuel temperature only. ρ
_{0}is the initial reactivity, whereas ρ_{C}(t) is time dependent reactivity inserted by reactor control system (e.g. by control rods or by boron dilution). This is the feedback equation. - The equations of the heat balance for fuel and coolant are interconnected via
*h(T*_{F}*– T*_{C}, which represents the heat transfer from the fuel into the coolant. In these equations,*)**m*and_{F}*m*are the mass of fuel and coolant in the core, respectively,_{C}*c*and_{pF}*c*are specific heat capacities of fuel and coolant,_{pC }is the heat transfer coefficient between fuel and coolant,*h**m*dotted is the coolant mass flow rate [kg/s] and T_{C}_{C}and T_{C,in}are the average and inlet coolant temperature, respectively.

To solve the point dynamics equations it is necessary to specify the initial conditions like in the case of point kinetics.

## Reactor Stability

At this point, we will discuss the reactor stability at power operation. At power operation, the neutron population is always large enough to generate heat. In fact, it is the main purpose of power reactors **to generate large amount of heat**. This causes the temperature of the system changes and material densities change as well (due to the **thermal expansion**).

These changes in reactivity are usually called the **reactivity feedbacks** and are characterized by **reactivity coefficients**. The reactivity feedbacks and their time constants are very important area of reactor design, because they determine the** stability of the reactor**.

In the previous article (Point Dynamics Equations) we have assumed a simplified feedback equation:

This equation expresses the dependence of the reactivity on various parameters. But in this case, there is a dependence on the **coolant** and the **fuel temperature** only. For PWRs, the temperature stability is of importance in overall stability, because most instabilities arise from the temperature instability. For illustration, we will use further simplified model, which assumes that there is only one temperature coefficient for fuel and moderator:

The response of a reactor to a change in temperature (i.e. the overall reactor stability) depends especially on the algebraic sign of **α**** _{T}**. Clearly, a reactor with

**negative**

**α**

**is**

_{T}**inherently stable**to changes in its temperature and thermal power while a reactor with

**positive**

**α**

**is**

_{T}**inherently unstable**. We will demonstrate the problem on following two examples.

**Positive reactivity feedback –** **α**_{T }**> 0**

_{T }

Assume **α**_{T }**> 0 (**temperature coefficient for fuel and moderator**). **If the temperature of the moderator is increased, positive reactivity is added to the core. This positive reactivity causes that reactor power further increases, which acts in the same direction as initial reactivity addition. Without compensation the reactivity of the system would increase and the thermal power would accelerate and increase as well (see figure).

On the other hand as the thermal power decreases, the reactor temperature decreases giving a further decrease in reactivity. This feedback would accelerate the initial decrease in thermal power and the reactor would shut down itself. In any case, the reactor power **does not stabilize itself**.

**Negative reactivity feedback –** **α**_{T }**< 0**

_{T }

The above situation is quite different when **α**_{T }**< 0. **In this case, if the temperature of the moderator is increased, **negative reactivity** is added to the core. This negative reactivity causes reactor power to decrease, which acts against any further increase in temperature or power. As the thermal power decreases, the power coefficient (which is also based on the sign of **α**** _{T}**) acts against this decrease and the reactor

**returns to the critical condition (steady-state)**. The

**reactor power stabilizes itself**. This effect is shown on the picture. Let assume all the changes are initiated by the changes in the

**core inlet temperature**.

At this point, it must be noted, that the temperature does not change uniformly throughout a reactor core. An increase in thermal power, for example, is reflected first by an increase in the temperature of the fuel, since this is the region, where most of the thermal power is generated. The coolant temperature and, in thermal reactors, the moderator temperature do not change until heat has been transferred from the fuel to the reactor coolant. The time for heat to be transferred to the moderator is usually measured in seconds (~5s).

Therefore it is necessary to specify the component whose temperature changes:

**Fuel temperature coefficient**– FTC or DTC is defined as the change in reactivity per degree change in the fuel temperature.

**α _{f} = ^{dρ}⁄_{dTf}**

The magnitude and sign (+ or -) of the **fuel temperature coefficient **is primarily a function of the fuel composition, especially the fuel enrichment. In power reactors, in which low enriched fuel (e.g. PWRs and BWRs require 3% – 5% of 235U) is used, **the Doppler coefficient is always negative**. In PWRs, the Doppler coefficient can range, for example, from **-5 pcm/°C to -2 pcm/°C**. This coefficient is of the highest importance in the **reactor stability**. The **fuel temperature coefficient** is generally considered to be even **more important **than the **moderator temperature coefficient**** (MTC)**. Especially in case of all **reactivity initiated accidents **(RIA), the fuel temperature coefficient will be **the first** and the most important feedback, that will compensate the inserted positive reactivity. The time constant for heating fuel is almost zero, therefore the fuel temperature coefficient is effective almost instantaneously. Therefore this coefficient is also called the **prompt temperature coefficient **because it causes an **immediate response** on changes in fuel temperature.

**Fuel Temperature Coefficient and Fuel Design**

In general, the **fuel temperature coefficient **is primarily a function of the **fuel enrichment**. There are **two phenomena** associated with Doppler effect. It results in **increased neutron capture** by fuel nuclei (both fissile and fissionable) in the resonance region, on the other hand it results in **increased neutron production** from fissile nuclei. Therefore DTC may be also positive. A net positive Doppler coefficient requires higher enrichments of fissile nuclei. Some calculations shows, that enrichments **higher than** **30%** are associated with slightly positive DTC, but this enrichment results in a harder neutron spectrum. Low enriched uranium oxide fuels usually provide a large negative DTC, because of the elastic scattering reaction of oxygen nuclei (soften spectrum). Another way to produce more negative DTC is to soften the neutron spectrum by addition of moderator nuclei directly into the fuel matrix. For example, the TRIGA reactor uses uranium zirconium hydride (UZrH) fuel, which has a large negative fuel temperature coefficient. The rise in temperature of the hydride increases the probability that a thermal neutron in the fuel element will gain energy from an excited state of an oscillating hydrogen atom in the lattice. As the neutrons gain energy from the ZrH, the thermal neutron spectrum in the fuel element shifts to a higher average energy (i.e., the spectrum is hardened). This spectrum hardening is used differently to produce the negative temperature coefficient.

In the United States, the Nuclear Regulatory Commission will not license a reactor unless **α**** _{Prompt}** (FTC) is negative. All licensed reactors are thereby assured of being inherently stable. This requirement is followed by many other countries.

**moderator temperature coefficient**is primarily a function of the

**moderator-to-fuel ratio**(

**N**). The

_{H2O}/N_{Fuel}ratio**moderator-to-fuel ratio**is the ratio of the number of moderator nuclei within the volume of a reactor core to the number of fuel nuclei. As the core temperature increases, fuel volume and number density remain essentially constant. The volume of moderator also remains constant, but the number density of moderator decreases with

**thermal expansion**. As the

**moderator temperature**increases the ratio of the moderating atoms (molecules of water) decreases as a result of the

**thermal expansion of water**(especially at 300°C; see: Density of Water). Its density simply and significantly decreases. This, in turn, causes a

**hardening of neutron spectrum**in the reactor core resulting in

**higher resonance absorption**(lower p). Decreasing density of the moderator causes that neutrons stay at a higher energy for a longer period, which increases the probability of non-fission capture of these neutrons. This process is one of three processes, which determine the

**moderator temperature coefficient (MTC)**. The second process is associated with the leakage probability of the neutrons and the third with the

**thermal utilization factor**.

**The moderator-to-fuel ratio **strongly influences especially:

**Resonance escape probability.**An increase in**moderator-to-fuel ratio causes**an increase in resonance escape probability. As more moderator molecules are added relative to the amount of fuel molecules, than it becomes easy for neutrons to slow down to thermal energies without encountering a resonance absorption at the resonance energies.**Thermal utilization factor.**An increase in**moderator-to-fuel ratio causes**a decrease in thermal utilization factor. The value of**the thermal utilization factor**is given by the ratio of the number of thermal neutrons absorbed in the fuel (**all nuclides**) to the number of thermal neutrons absorbed in**all the material**that makes up the core.- Thermal and fast non-leakage probability. An increase in
**moderator-to-fuel ratio causes**a decrease in migration length, which in turn causes an increase in non-leakage probability.

As can be seen from the figure, at low **moderator-to-fuel ratios **the product of all the six factors (k_{eff}) is small because the resonance escape probability is small. At optimal value of **moderator-to-fuel ratio, **k_{eff} reaches its maximum value. This is the case of so called “optimal moderation”. At large ratios, k_{eff} is again small because **the thermal utilization factor** is small.

**Under-moderated vs. Over-moderated Reactor**

From the **moderator-to-fuel ratio** point of view, any multiplying system can be designed as:

**Under-moderated**. Under-moderation means that there is**less than optimum**amount of**moderator**between fuel plates or fuel rods. An increase in moderator temperature and voids decreases k_{eff}of the system and inserts negative reactivity. An under-moderated core would create a negative temperature and void feedback required for a**stable system**.

**Over-moderated**. Over-moderation means that there is**higher than optimum**amount of**moderator**between fuel plates or fuel rods. An increase in moderator temperature and voids increases k_{eff}of the system and inserts positive reactivity. An over-moderated core would create a positive temperature and void feedback. It will result in an**unstable system**, unless another negative feedback mechanism (e.g. the Doppler broadening) overrides the positive effect.

Reactor engineers must balance the composite effects of moderator density, fuel temperature, and other phenomena to ensure system stability under all operating conditions. Most of light water reactors are therefore designed as so called **under-moderated** and the **neutron flux spectrum** is slightly **harder** (the moderation is slightly insufficient) than in an optimum case. But this design provides important safety feature. An increase in the moderator temperature results in negative reactivity which tends to make the reactor **self-regulating**. It must be added, the overall feedback must be negative, but** local positive coefficients** exists in areas with **large water gaps** that are over-moderated such as near control rods guide tubes.

Another phenomenon associated with under-moderated core is called the **neutron flux trap effect**. This effect causes an increase in local power generation due better thermalisation of neutrons in areas with large water gaps (between fuel assemblies or when fuel assembly bow phenomenon is present). Note that “flux traps” are a standard feature of most modern test reactors because of the desire to obtain high thermal neutron fluxes for the irradiation of materials, but basically it can occur also in PWRs.

On the other hand, also **under-moderation** has its **limits**. In general, it causes a decrease in overall k_{eff}, therefore more fissile material is needed to ensure criticality of the core. Moreover, there is also a limit on the minimal value of MTC (most negative). It is due to the fact the negative temperature feedback acts also against decrease in the moderator temperature. Consider what happens when moderator temperature is decreased **quickly**, as in the case of the **main steamline break** (MSLB – standard initiating event for PWRs). The steamline break causes the steam pressure, the saturation temperature in the steam generators to fall rapidly. As a result of falling saturation temperature in the steam generators the **moderator temperature** will rapidly decrease. The rapid moderator temperature drop causes a positive reactivity insertion. The amount of reactivity inserted depends also on a magnitude of the MTC and therefore it must be limited. Typical values for lower limit is MTC = -80 pcm/°C, but it is a plant specific value limited in technical specifications.

### Examples: Reactor Stability

**negative MTC**is favorable operational characteristics also during

**power changes**. At normal operation there is an exact

**energy balance**between the primary circuit and secondary circuit. Therefore when the operator decreases the load on the turbine (e.g. due to a grid requirement), the steam demand decreases (see the initial electrical output decrease at the picture). At this moment, the reactor will produce more heat than the steam turbine can consume. This

**disbalance**causes the steam pressure, the saturation temperature in the steam generators to increase (see II. pressure at the picture). As a result of increasing saturation temperature in the steam generators the

**moderator temperature**will simply increase (see inlet temperature). Increasing the temperature of the moderator adds

**negative reactivity**, which reduces reactor power (without any operator intervention). As can be seen, to a certain extent the reactor is

**self-regulating**and the reactor power may be controlled via the steam turbine and via grid requirements. This feature is limited, because also the range of allowable inlet temperatures is limited. It is power plant specific, but in general, power changes of the order of units of % are common.

**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 boron dilution). 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%**

**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 **100%** of rated power and that the plant operator wants to decrease power to **75%** of rated power. The reactor operator must first bring the reactor subcritical by insertion of a negative reactivity (e.g. by control rod insertion or boric acid addition). As the thermal power decreases, moderator temperature and fuel temperature decrease as well, causing a positive reactivity effect (from the power coefficient) and the reactor returns to the critical condition. In order to keep the power to be decreasing, **negative 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.

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**

**100% → ↓ 20 steps or ↑ 18 ppm of boric acid within 10 minutes → 90%→ next ↓ 20 steps or ↑ 18 ppm within 10 minutes → 80% → final ↓ 10 steps or ↑ 9 ppm within 5 minutes→ 75%**

**Nuclear and Reactor Physics:**

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- U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.

**Advanced Reactor Physics:**

- K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
- K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.
- D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2.
- E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.

### See above:

Contents

- Reactor Dynamics
- Prompt and Delayed Neutrons
- Key Characteristics of Prompt Neutrons
- Key Characteristics of Delayed Neutrons
- Point Kinetics Equations
- Derivation of Simple Point Kinetics Equation
- Simple Point Kinetics Equation without Delayed Neutrons
- Simple Point Kinetics Equation with Delayed Neutrons
- Point Kinetics Equations
- Inhour Equation
- Special Cases of Inhour Equation
- Reactivity Pulse – Impulse Characteristics
- Oscillation of Reactivity – Frequency Characteristics
- Approximate Solution of Point Kinetics Equations
- Experimental Methods of Reactivity Determination
- Reactivity
- Inverse Reactor Kinetics – Reactimeter
- Reactivity Coefficients – Reactivity Feedbacks
- How negative feedback acts against power excursion
- Feedback Delay – Time Constants
- Point Dynamics Equations
- Reactor Stability
- Reactor Physics