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
- Example: Change in the moderator temperature.
- Examples: Change in the reactor power
- Feedback Delay – Time Constants
- Point Dynamics Equations
- Reactor Stability
- Positive reactivity feedback – αT > 0
- Negative reactivity feedback – αT < 0
- Examples: Reactor Stability
- Test your Knowledge – Reactor Dynamics

## 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.

Source: JANIS (Java-based Nuclear Data Information Software); The JEFF-3.1.1 Nuclear Data Library

## 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.

## 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****.**

## 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.

## 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:

## 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.

## Special Cases of Inhour Equation

## 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

## 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:

## 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.

## 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)

## 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**).

**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

**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).

## Examples: Change in the reactor power

**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:

## Examples: Reactor Stability

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