{"id":13997,"date":"2017-03-10T17:36:45","date_gmt":"2017-03-10T17:36:45","guid":{"rendered":"http:\/\/sitepourvtc.com\/?page_id=13997"},"modified":"2022-10-21T16:57:17","modified_gmt":"2022-10-21T16:57:17","slug":"reactivity-coefficients-reactivity-feedbacks","status":"publish","type":"page","link":"https:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/reactivity-coefficients-reactivity-feedbacks\/","title":{"rendered":"Reactivity Coefficients – Reactivity Feedbacks"},"content":{"rendered":"
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Reactivity feedbacks<\/strong> are inherent feedbacks that determine the\u00a0stability of the reactor<\/strong>. Reactivity coefficients<\/strong> characterize these feedbacks. Reactivity coefficients are the amount that the reactivity will change for a given change in the parameter. Reactor design must assure that under all operating conditions, the temperature feedback will be negative<\/strong>.\n

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

“The 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.<\/p>\n<\/div><\/div>\n

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

However, in an operating power reactor,<\/strong> the neutron population is always large enough to generate heat. It is the main purpose of power reactors to generate a large amount of heat<\/strong>. This causes the system’s temperature to change and material densities to change as well (due to the thermal expansion<\/strong>).<\/p><\/div><\/div>

<\/iframe><\/div>Demonstration of\u00a0the prompt negative temperature coefficient<\/strong> at the TRIGA reactor<\/strong>.\u00a0A 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\u00a0allows TRIGA reactors to operate safely<\/strong> during either steady-state<\/strong> or transient conditions<\/strong>.\n

Source: Youtube<\/a><\/p>\n

See also: General Atomics\u00a0– TRIGA<\/a><\/p><\/div><\/div>

Because macroscopic cross-sections<\/a> are proportional to densities<\/a> and temperatures, neutron flux spectrum<\/strong><\/a> depends also on the density of moderator<\/a>, these changes in turn will produce some changes in reactivity. These changes in reactivity are usually called reactivity feedbacks<\/strong> and are characterized by reactivity coefficients<\/strong>. This is a very important area of reactor design because the reactivity feedbacks influence the stability of the reactor<\/strong>. For example, reactor design must ensure that the temperature feedback will be negative <\/strong>under all operating conditions.
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Example:\u00a0Change in the moderator temperature.<\/h2>\n

Negative feedback<\/strong> 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 stabilizes itself. In terms of multiplication factor, this effect is caused by significant changes in the resonance escape probability<\/a> and total neutron leakage<\/a>\u00a0(or in the thermal utilization factor<\/a> when the chemical shim<\/a> is used).<\/p>\n

<\/span>Resonance escape probability<\/div>
\u2191TM<\/sub>\u00a0\u21d2\u00a0\u2193keff<\/sub> = \u03b7.\u03b5. \u00a0\u2193p \u00a0. \u2191f . \u00a0\u2193Pf \u00a0<\/sub>. \u00a0\u2193Pt\u00a0<\/sub> (BOC)<\/strong><\/p>\n

\u2191TM<\/sub>\u00a0\u21d2\u00a0\u2193keff<\/sub> = \u03b7.\u03b5. \u00a0\u2193p \u00a0.f. \u00a0\u2193Pf \u00a0<\/sub>. \u00a0\u2193Pt\u00a0<\/sub> (EOC)<\/strong><\/p>\n

<\/span>Thermal utilization factor<\/div>
\u2191TM<\/sub>\u00a0\u21d2\u00a0\u2193keff<\/sub> = \u03b7.\u03b5. \u00a0\u2193p \u00a0. \u2191f . \u00a0\u2193Pf \u00a0<\/sub>. \u00a0\u2193Pt\u00a0<\/sub> (BOC)<\/strong><\/p>\n

\u2191TM<\/sub>\u00a0\u21d2\u00a0\u2193keff<\/sub> = \u03b7.\u03b5. \u00a0\u2193p \u00a0.f. \u00a0\u2193Pf \u00a0<\/sub>. \u00a0\u2193Pt\u00a0<\/sub> (EOC)<\/strong><\/p>\n

<\/span>Neutron leakage<\/div>
\u2191TM<\/sub>\u00a0\u21d2\u00a0\u2193keff<\/sub> = \u03b7.\u03b5. \u00a0\u2193p \u00a0. \u2191f . \u00a0\u2193Pf \u00a0<\/sub>. \u00a0\u2193Pt\u00a0<\/sub> (BOC)<\/strong><\/p>\n

\u2191TM<\/sub>\u00a0\u21d2\u00a0\u2193keff<\/sub> = \u03b7.\u03b5. \u00a0\u2193p \u00a0.f. \u00a0\u2193Pf \u00a0<\/sub>. \u00a0\u2193Pt\u00a0<\/sub> (EOC)<\/strong><\/p>\n

Change of\u00a0the neutron leakage.\u00a0<\/strong>Since both (Pf<\/sub> and Pt<\/sub><\/strong>) are affected by a change in moderator temperature<\/strong> in a heterogeneous water-moderated reactor and the directions of the feedbacks for both negative, the resulting total non-leakage probability<\/strong> is also sensitive to the change in the moderator temperature. As a result, an increase in the moderator temperature<\/strong> causes that the probability of leakage to increase<\/strong>. In the case of the fast neutron leakage,\u00a0<\/strong>the moderator temperature influences macroscopic cross-sections<\/a> for elastic scattering reaction<\/a> (\u03a3s<\/sub>=\u03c3s<\/sub>.NH2O<\/sub>) due to the thermal expansion of water, which increases the moderation length. This, in turn, causes an increase in the leakage of fast neutrons.<\/p>\n

Thermal utilization factor. <\/strong>The impact on the thermal utilization factor<\/a> depends strongly on the amount of boron<\/a> that is diluted in the primary coolant (chemical shim<\/a>).\u00a0<\/strong>As the moderator temperature increases, the density of water decreases due to the thermal expansion<\/strong> of water. But along with the moderator also\u00a0boric acid is expanded<\/strong> out of the core. Since boric acid is a neutron poison, expanding out of the core, positive reactivity is added. The positive reactivity addition due to the expansion of boron out of the core offsets the negative reactivity addition due to the expansion of the moderator out of the core. Obviously, this effect is significant at the beginning of the cycle<\/strong> (BOC) and gradually loses its significance as the boron concentration decreases.<\/div><\/div>

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