{"id":18293,"date":"2018-07-29T07:14:40","date_gmt":"2018-07-29T07:14:40","guid":{"rendered":"http:\/\/sitepourvtc.com\/?page_id=18293"},"modified":"2023-01-31T10:06:13","modified_gmt":"2023-01-31T10:06:13","slug":"reactor-stability","status":"publish","type":"page","link":"https:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/reactor-dynamics\/reactor-stability\/","title":{"rendered":"Reactor Stability"},"content":{"rendered":"
The response of a reactor<\/a> to a change in temperature (i.e., the overall reactor stability) depends especially on the algebraic sign of \u03b1<\/strong>T<\/sub><\/strong>. A reactor with negative<\/strong> \u03b1<\/strong>T<\/sub><\/strong> is inherently stable<\/strong> to changes in its temperature and thermal power, while a reactor with positive<\/strong> \u03b1<\/strong>T<\/sub><\/strong> is inherently unstable<\/strong>.<\/p>\n

According to 10 CFR Part 50<\/a>; Criterion 11:<\/p>\n

\u201cThe reactor core and associated coolant systems shall be designed so that in power operating range, the net effect of the prompt inherent nuclear feedback characteristics tends to compensate for a rapid increase in reactivity.\u201d<\/p>\n<\/div><\/div>\n

At this point, we will discuss the reactor stability at power operation. The neutron population is always large enough to generate heat at power operation. The main purpose of power reactors is to generate a large amount of heat, and this<\/strong> causes the temperature of the system to change, and material densities change as well (due to the thermal expansion<\/strong>).<\/p>\n

\"reactivity<\/a>
Reactivity and reactor power as a function of time and various temperature feedbacks.<\/figcaption><\/figure>\n

These changes in reactivity are usually called reactivity feedbacks<\/strong><\/a> and are characterized by reactivity coefficients<\/strong>. The reactivity feedbacks and their time constants are a very important area of reactor design because they determine the stability of the reactor<\/strong>.<\/p>\n

In the previous article (Point Dynamics Equations<\/a>), we have assumed a simplified feedback equation:<\/p>\n

\"equation<\/a><\/p>\n

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

\"temperature<\/a><\/p>\n

The response of a reactor<\/a> to a change in temperature (i.e., the overall reactor stability) depends especially on the algebraic sign of \u03b1<\/strong>T<\/sub><\/strong>. A reactor with negative<\/strong> \u03b1<\/strong>T<\/sub><\/strong> is inherently stable<\/strong> to changes in its temperature and thermal power, while a reactor with positive<\/strong> \u03b1<\/strong>T<\/sub><\/strong> is inherently unstable<\/strong>. We will demonstrate the problem in the following two examples.<\/p>\n

Positive reactivity feedback – \u03b1T <\/sub>> 0<\/h2>\n

Assume \u03b1<\/strong>T <\/sub><\/strong>> 0 (<\/strong>temperature coefficient for fuel and moderator). <\/strong>If the temperature of the moderator is increased, positive reactivity is added to the core. This positive reactivity causes reactor power to further increase, 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).<\/p>\n

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<\/strong>.<\/p>\n

Negative reactivity feedback – \u03b1T <\/sub>< 0<\/h2>\n

\"moderator<\/a><\/p>\n

The above situation is quite different when \u03b1<\/strong>T <\/sub><\/strong>< 0. <\/strong>In this case, if the temperature of the moderator is increased, negative reactivity<\/strong> 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 \u03b1<\/strong>T<\/sub><\/strong>) acts against this decrease, and the reactor returns to the critical condition<\/a> (steady-state)<\/strong>. The reactor power stabilizes itself<\/strong>. This effect is shown in the picture. Let\u2019s assume all the changes are initiated by the changes in the core inlet temperature<\/strong>.<\/p>\n

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 does not change until the heat has been transferred from the fuel to the reactor coolant. The time for heat to be transferred to the moderator<\/a> is usually measured in seconds (~5s).<\/p>\n

Therefore it is necessary to specify the component whose temperature changes:<\/p>\n

<\/span>Fuel Temperature Coefficient and Fuel Design<\/div>
Fuel temperature coefficient<\/strong><\/a> \u2013 FTC or DTC is defined as the change in reactivity per degree change in the fuel temperature.<\/p>\n

\u03b1f<\/sub> = d\u03c1<\/sup>\u2044dTf<\/sub><\/span><\/strong><\/p>\n

The magnitude and sign (+ or -) of the fuel temperature coefficient <\/strong>is primarily a function of the fuel composition, especially the fuel enrichment. In power reactors, in which low enriched fuel (e.g., \u00a0PWRs and BWRs require 3% \u2013 5% of 235U<\/a>) is used, the Doppler coefficient is always negative<\/strong>. In PWRs<\/a>, the Doppler coefficient can range, for example, from -5 pcm\/\u00b0C to -2 pcm\/\u00b0C<\/strong>. This coefficient is of the highest importance in reactor stability<\/strong>. The fuel temperature coefficient<\/strong> is generally considered even more important <\/strong>than the moderator temperature coefficient<\/strong><\/a> (MTC)<\/strong>. Especially in the case of all reactivity-initiated accidents <\/strong>(RIA), the fuel temperature coefficient will be the first<\/strong> and the most important feedback that will compensate for the inserted positive reactivity. The time constant for heating fuel is almost zero. Therefore, the fuel temperature coefficient is almost instantaneously. Therefore this coefficient is also called the prompt temperature coefficient <\/strong>because it causes an immediate response<\/strong> to changes in fuel temperature.<\/p>\n

Fuel Temperature Coefficient and Fuel Design<\/strong><\/p>\n

In general, the fuel temperature coefficient <\/strong>is primarily a function of the fuel enrichment<\/strong>. There are two phenomena<\/strong> associated with the Doppler effect. It increases\u00a0neutron capture<\/strong> by fuel nuclei (both fissile and fissionable) in the resonance region. On the other hand, it results in increased neutron production<\/strong>\u00a0from fissile nuclei<\/a>. Therefore DTC may also be positive. A net positive Doppler coefficient requires higher enrichments of fissile nuclei. Some calculations show that enrichments higher than<\/strong> 30%<\/strong> 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<\/a> of oxygen nuclei (soften spectrum). Another way to produce more negative DTC is to soften the neutron spectrum by adding moderator nuclei directly into the fuel matrix. For example, the TRIGA reactor<\/a> 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 higher average energy (i.e., the spectrum is hardened). This spectrum hardening is used differently to produce the negative temperature coefficient.<\/p>\n

In the United States, the Nuclear Regulatory Commission will not license a reactor unless \u03b1<\/strong>Prompt<\/sub><\/strong> (FTC) is negative. All licensed reactors are thereby assured of being inherently stable. This requirement is followed by many other countries.<\/div><\/div>

<\/span>Moderator Temperature Coefficient and Reactor Design<\/div>
\"moderator-to-fuel<\/a>As was written, the moderator temperature coefficient<\/strong><\/a> is primarily a function of the moderator-to-fuel ratio <\/strong>(NH2O<\/sub>\/NFuel<\/sub> ratio<\/strong>). The moderator-to-fuel ratio <\/strong>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<\/a> of moderator decreases with thermal expansion<\/strong>. As the moderator temperature<\/strong> increases, the ratio of the moderating atoms (molecules of water) decreases due to the thermal expansion of water <\/strong>(especially at 300\u00b0C; see: Density of Water<\/a>). Its density simply and significantly decreases. This, in turn, causes hardening of neutron spectrum<\/strong> in the reactor core resulting in higher resonance absorption<\/strong> (lower p). The decreasing density of the moderator causes neutrons<\/a> to 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 that determine the moderator temperature coefficient (MTC)<\/strong>. The second process is associated with the leakage probability<\/a> of the neutrons and the third with the thermal utilization factor<\/strong>.<\/p>\n

The moderator-to-fuel ratio <\/strong>strongly influences especially:<\/p>\n