## Reactor Criticality

In preceding chapters, the multiplication factors have been analysed in order to understand a **steady-state condition**. Also the operational changes that influence **the criticality** of the reactor were described. It must be noted, nuclear reactors are not always critical. In order to start it up or to shut it down, the **k _{eff}** must be changed for a certain period, which in turn changes the neutron population in the reactor core. The study of behavior of the neutron population in a

**noncritical reactor**is called

**reactor kinetics**. From this point of view, the state of a reactor can be distinguished by following criteria.

**The basic classification** of states of a reactor is according to the **multiplication factor** **as eigenvalue** which is a measure of the change in the fission neutron population from one neutron generation to the subsequent generation.

**k**. If the multiplication factor for a multiplying system is_{eff}< 1**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**. If the multiplication factor for a multiplying system is_{eff}= 1**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**. If the multiplication factor for a multiplying system is_{eff}> 1**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**.

The definitions described above are fully applicable to a reactor at **zero power level**, that is at such power level in which **all thermal considerations** are not important to the chain reaction (let say **from zero power to 1% of rated power**).

**The effective multiplication factor**(

**k**) may be expressed mathematically in terms of the infinite multiplication factor (k

_{eff}_{∞}) and two additional factors which account for

**neutron leakage**during neutron thermalisation (

**fast non-leakage probability**) and neutron leakage during neutron diffusion (

**thermal non-leakage probability**) by following equation, usually known as the

**six factor formula**:

**k _{eff} = k_{∞} . P_{f} . P_{t}**

**The simplest equation governing the neutron kinetics of the system with delayed neutrons is**

**the point kinetics equation**. This equation states that the time change of the neutron population is equal to the

**excess of neutron production**(by fission)

**minus neutron loss**by absorption

**in one mean generation time with delayed neutrons (ld)**. The role of

**ld**is evident.

**Longer lifetimes give simply slower 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 point kinetics equation with delayed neutrons (similarly to the case without delayed neutrons):**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. 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.

Reactors with such a kinetics would be quite **easy to control**. From this point of view it may seem that reactor control will be a quite boring affair. It will not! The presence of delayed neutrons entails many many specific phenomena, that will be described in later chapters.

**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).

**the neutron population**) in the core at time zero is 1000 and k

_{∞}= 1.001 (~100 pcm).

Calculate the number of neutrons after 100 generations. Let say, the mean generation time is ~0.1s.

**Solution:**

To calculate the neutron population after 100 neutron generations, we use following equation:

N_{n}=N_{0}. (k_{∞})^{n}

N_{1}=N_{0}.1.001 = **1001 neutrons after one generation**

N_{2}=N_{0}.1.001.1.001 = **1002 neutrons after two generations**

N_{3}=N_{0}.1.001.1.001.1.001 = **1003 neutrons after three generations**

.

.

N_{50}=N_{0}. (k_{∞})^{50} = **1051 neutrons after fifty generations**.

.

.

N_{100}=N_{0}. (k_{∞})^{100} = **1105 neutrons after hundred generations.**

If we consider the mean generation time to be ~0.1s, so the increase from 1000 neutrons the 1105 neutrons occurs within 10 seconds.

See also: Neutron Generation – Neutron Population

## Criticality of a Power Reactor – Power Defect

**For power reactors**at power conditions the reactor can behave

**differently**as a result of the

**presence of reactivity feedbacks**. Power reactors are initially started up from

**hot standby mode**(subcritical state at 0% of rated power) to

**power operation mode**(100% of rated power) by withdrawing control rods and by boron dilution from the primary coolant. During the reactor startup and up to about 1% of rated power, the reactor kinetics is

**exponential**as in zero power reactor. This is due to the fact all temperature reactivity effects are minimal.

On the other hand, during further power increase from about 1% up to 100% of rated power, **the temperature reactivity effects play very important role**. As the neutron population increases, the fuel and the moderator increase its **temperature**, which results in **decrease in reactivity** of the reactor (almost all reactors are designed to have the **temperature coefficients negative**).

See also: Operational factors that affect the multiplication in PWRs

**The negative reactivity coefficient** acts against the initial positive reactivity insertion and this positive reactivity is **offset** by negative reactivity from **temperature feedbacks**. 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 on the power level 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 is known as **the power defect**. **The power defects** for PWRs, graphite-moderated reactors, and sodium-cooled fast reactors are:

- about
**2500pcm**for PWRs, - about
**800pcm**for graphite-moderated reactors - about
**500pcm**for sodium-cooled fast reactors

**The power defects** slightly depend on the fuel burnup, because they are determined by the **power coefficient** which depends on the fuel burnup. **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**.

It is logical, as power defects act against power increase, they act also **against power decrease**. When reactor power is decreased **quickly**, as in the case of **reactor trip**, power defect causes a positive reactivity insertion, and the initial rod insertion must be sufficient to make the reactor **safe subcritical**. It is obvious, if the power defect for **PWRs** is about **2500pcm** (about 6 βeff), the control rods must weigh **more than 2500pcm** to achieve the **subcritical condition**. To ensure the **safe subcritical condition**, the control rods must weigh more than 2500pcm plus value of **SDM** (SHUTDOWN MARGIN). The total weigh of control rods is design specific, but, for example, it may reach about 6000pcm. To ensure that the control rods can **safe shut down the reactor**, they must be maintained above a minimum rod height (rods insertion limits).

## Prompt Criticality

**The basic classification**of states of a reactor may be in some case

**insufficient**and a finer classification is needed. The finer classification takes into account the two groups of neutrons that are produced in fission.

It is known the fission neutrons are of importance in any chain-reacting system. The fission of fissile 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.

The fact the prompt neutrons are emitted within **10 ^{-14}** and have significant shorter mean generation time (

**~ 10**) than delayed neutrons (

^{-5}s**~0.1s**) have crucial consequences. The period of ~

**10**is very short and causes very very fast response of the reactor power in case of

^{-5}s**prompt criticality**. The state of reactor, when the chain reaction is self-sustained only by prompt neutrons, is known as the

**prompt critical state**. This state of the reactor is very

**unstable,**because one neutron generation takes only ~

**10**Therefore nuclear reactors

^{-5}s.**must operate**in the

**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.

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

**k**, 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_{eff}> 1; ρ ≥ β_{eff}**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**, where the reactivity of a reactor is_{eff}> 1; 0 < ρ < β_{eff}**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**, where the reactivity of a reactor is_{eff}= 1; ρ = 0**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**, 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 s_{eff}< 1; ρ < 0**ubcritical multiplication**.

**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**, is the same fraction at thermal energies.

_{eff}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%.

**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.

**Nuclear and Reactor Physics:**

- J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
- J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
- W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
- Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
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- G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
- Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
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**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.
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- E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.

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