## 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 }

**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%**

### Flow Instabilities in BWRs

**Coupled Neutronic-Thermohydraulic Instability**

See also: Flow Instability

The dominant type of instabilities in commercial BWRs is the coupled neutronic-thermohydraulic instability (also known as the **reactivity instability**). The power generation in BWRs is directly related to the fuel **neutron flux**, which is strongly related to the average **void fraction** in the core channels through. This effect is known as the** reactivity feedback**. The reactivity feedback caused by changes in void fraction (void coefficient) is delayed as the voids travel upward through the fuel channel. In some cases the delay may be long enough and the **void feedback** may be strong enough that the reactor configuration becomes unstable. In this case the neutron flux may oscillate.

**Nuclear and Reactor Physics:**

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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.
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