{"id":13510,"date":"2017-01-03T09:11:38","date_gmt":"2017-01-03T09:11:38","guid":{"rendered":"http:\/\/sitepourvtc.com\/?page_id=13510"},"modified":"2022-10-19T16:37:43","modified_gmt":"2022-10-19T16:37:43","slug":"neutron-flux-neutron-intensity","status":"publish","type":"page","link":"https:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/neutron-flux-neutron-intensity\/","title":{"rendered":"Neutron Flux Density – Neutron Intensity"},"content":{"rendered":"
The neutron flux density<\/strong>, \u0424<\/strong>, is the number of neutrons crossing through some arbitrary cross-sectional unit area in all directions<\/strong> per unit time. It is a scalar quantity,<\/strong>\u00a0and it can be calculated as the neutron density (n)<\/strong> multiplied by neutron velocity (v)<\/strong>.<\/div><\/div>\n

The section on the neutron cross-section<\/a>\u00a0determined the probability of a neutron<\/a> undergoing a specific neutron-nuclear reaction<\/a>. It was determined the mean free path<\/a> of neutrons in the material under specific conditions. These parameters influence the criticality of the reactor core<\/strong>. In other words, we do not know anything about the power level<\/strong> of the reactor core<\/a>. If we want to know the reactor core’s reaction rate<\/strong> or thermal power<\/strong>, it is necessary to know how many neutrons<\/strong> are traveling through the material.<\/p>\n

\"thermal<\/a>
Distribution of kinetic energies of neutrons in the thermal reactor and the fast neutrons reactor. The fission neutrons (fast-flux) in the thermal reactor are immediately slowed down to the thermal energies via neutron moderation.<\/figcaption><\/figure>\n

It is convenient to consider the neutron density<\/strong>, the number of neutrons existing in one cubic centimeter. The symbol n<\/strong> represents the neutron density\u00a0with units of neutrons\/cm3<\/sup>. In reactor physics, the neutron flux<\/strong> is more likely used because it expresses better the total path length<\/strong> covered by all neutrons<\/strong>. Their velocity determines the total distance these neutrons can travel each second. The neutron flux density<\/strong> value is calculated as the neutron density (n)<\/strong> multiplied by neutron velocity (v)<\/strong>.<\/p>\n

\u0424 = n.v<\/strong><\/p>\n

where:
\n\u0424 – neutron flux (neutrons.cm-2<\/sup>.s-1<\/sup>)<\/strong>
\n n – neutron density (neutrons.cm-3<\/sup>)<\/strong>
\n v – neutron velocity (cm.s-1<\/sup>)<\/strong><\/p>\n

The neutron flux<\/strong>, the number of neutrons crossing through some arbitrary cross-sectional unit area in all directions<\/strong> per unit time, is a scalar quantity<\/strong>. Therefore it is also known as the scalar flux<\/strong>. The expression \u0424(E).dE<\/strong> is the total distance traveled during one second by all neutrons with energies between E and dE located in 1 cm3<\/sup>.<\/p>\n

The connection to the reaction rate<\/strong>, respectively, the reactor power is obvious. Knowledge of the neutron flux<\/strong> (the total path length<\/strong> of all the neutrons in a cubic centimeter in a second) and the macroscopic cross-sections<\/strong><\/a> (the probability of having an interaction per centimeter path length<\/strong>) allows us to compute the rate of interactions (e.g.,, rate of fission reactions). The reaction rate<\/strong> (the number of interactions taking place in that cubic centimeter in one second) is then given by multiplying them together:<\/p>\n

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

where:
\n\u0424 – neutron flux (neutrons.cm-2<\/sup>.s-1<\/sup>)<\/strong>
\n\u03c3 – microscopic cross-section (cm2<\/sup>)<\/strong>
\nN – atomic number density (atoms.cm-3<\/sup>)<\/strong><\/p>\n

\n

Neutron Flux and Intensity – Examples<\/h2>\n

We have to distinguish between the neutron flux<\/strong> and the neutron intensity<\/strong>. Although both physical quantities have the same units<\/strong>, namely, neutrons.cm-2<\/sup>.s-1<\/sup>, their physical interpretations are different. In contrast to the neutron flux, the neutron intensity is the number of neutrons crossing through some arbitrary cross-sectional unit area in a single direction<\/strong> per unit time (a surface is perpendicular to the direction of the beam). The neutron intensity<\/strong> is a vector quantity<\/strong>.<\/p>\n

<\/span>Example - Neutron flux in a typical thermal reactor core<\/div>
A typical thermal reactor<\/strong> contains about 100 tons<\/strong> of uranium<\/a> with an average enrichment of 2%<\/strong> (do not confuse it with the enrichment of the\u00a0fresh fuel<\/strong>). If the reactor power is 3000MWth<\/sub>, determine the reaction rate<\/strong> and the average core thermal flux<\/strong>.<\/p>\n

Solution:<\/strong><\/p>\n

Multiplying the reaction rate per unit volume (RR = \u0424 . \u03a3) by the total volume of the core (V) gives us the total number of reactions<\/strong> occurring in the reactor core<\/a> per unit time. But we also know the amount of energy released per one fission reaction<\/a> to be about 200 MeV\/fission<\/strong>. Now, it is possible to determine the energy release rate<\/b> (power) due to the fission reaction. The following equation gives it:<\/p>\n

P = \u0424 . \u03a3f<\/sub> . Er<\/sub> . V = \u0424 . NU235<\/sub> . \u03c3f<\/sub>235<\/sup> . Er<\/sub> . V<\/strong><\/p>\n

where:
\nP – reactor power (MeV.s-1<\/sup>)<\/strong>
\n \u0424 – neutron flux (neutrons.cm-2<\/sup>.s-1<\/sup>)<\/strong>
\n \u03c3 – microscopic cross section (cm2<\/sup>)<\/strong>
\n N – atomic number density (atoms.cm-3<\/sup>)<\/strong>
\n Er – the average recoverable energy per fission (MeV \/ fission)<\/strong>
\n V – total volume of the core (m3<\/sup>)<\/strong><\/p>\n

The amount of fissile<\/a> 235<\/sup>U<\/a> per the volume of the reactor core.<\/p>\n

m235<\/sub>\u00a0[g\/core] = 100 [metric tons] x 0.02 [g of 235<\/sup>U\u00a0\/ g of U] . 106<\/sup>\u00a0[g\/metric ton]\n= 2 x 106<\/sup> grams of 235<\/sup>U<\/strong>\u00a0per the volume of the reactor core<\/p>\n

The atomic number density <\/a>of 235<\/sup>U\u00a0in the volume of the reactor core:<\/p>\n

N235<\/sub> . V = m235<\/sub> . NA<\/sub> \/ M235<\/sub>
\n= 2 x 106<\/sup> [g 235 \/ core] x 6.022 x 1023<\/sup>\u00a0[atoms\/mol] \/ 235 [g\/mol]\n= 5.13 x 1027<\/sup> atoms \/ core<\/strong>
\nThe microscopic fission cross-section of 235<\/sup>U (for thermal neutrons<\/a>):<\/p>\n

\u03c3f<\/sub>235<\/sup> = 585 barns<\/strong><\/p>\n

The average recoverable energy per 235<\/sup>U\u00a0fission:<\/p>\n

Er<\/sub> = 200.7 MeV\/fission<\/strong><\/p>\n

\"Neutron<\/a><\/p>\n<\/div><\/div>\n

<\/span>Example - Neutron flux in a MOX fueled thermal reactor core<\/div>
Mixed oxide fuel<\/strong>, commonly referred to as MOX fuel<\/strong>, is nuclear fuel<\/a> that contains more than one oxide of fissile material<\/a>. MOX fuels usually consist of plutonium<\/a> blended with natural uranium<\/a>. For simplicity and purposes of this example, it will be assumed that the reactor core<\/a> contains only MOX fuel (100% MOX) and that the averaged percentage of plutonium<\/strong> (averaged over all the used fuel assemblies – not all FAs are fresh FAs<\/strong>) in MOX fuel is equal to 4%<\/strong>. Note that the initial percentage of plutonium in fresh MOX fuel is around\u00a0~7%<\/strong>.<\/p>\n

Moreover, it will be assumed. The recycled plutonium contains only the fissile 239<\/sup>Pu<\/a>. In reality, MOX fuel always contains significant amounts of higher isotopes – 240<\/sup>Pu<\/a>, 241<\/sup>Pu<\/a>, and 242<\/sup>Pu. The presence of these amounts will be neglected in this example<\/strong>.<\/p>\n

A typical thermal reactor<\/strong> contains about 100 tons of fuel<\/strong> (HM – heavy metal). If the reactor power is 3000MWth, determine the reaction rate<\/strong> and the average core thermal flux<\/strong>.<\/p>\n

Solution:<\/strong><\/p>\n

Multiplying the reaction rate per unit volume (RR = \u0424 . \u03a3) by the total volume of the core (V) gives us the total number of reactions<\/strong> occurring in the reactor core per unit time. But we also know the amount of energy released per one fission reaction<\/a>\u00a0is about 207 MeV\/fission<\/strong>. Now, it is possible to determine the energy release rate<\/strong> (power) due to the fission reaction. The following equation gives it:<\/p>\n

P = \u0424 . \u03a3f<\/sub> . Er<\/sub> . V = \u0424 . NPu239<\/sub>\u00a0. \u03c3f<\/sub>239<\/sup>\u00a0. Er<\/sub> . V<\/strong><\/p>\n

where:
\nP – reactor power (MeV.s-1<\/sup>)<\/strong>
\n \u0424 – neutron flux (neutrons.cm-2<\/sup>.s-1<\/sup>)<\/strong>
\n \u03c3 – microscopic cross section (cm2<\/sup>)<\/strong>
\n N – atomic number density (atoms.cm-3<\/sup>)<\/strong>
\n Er – the average recoverable energy per fission (MeV \/ fission)<\/strong>
\n V – total volume of the core (m3<\/sup>)<\/strong><\/p>\n

The amount of fissile<\/a>\u00a0239<\/sup>Pu\u00a0per the volume of the reactor core.<\/p>\n

m239<\/sub>\u00a0[g\/core] = 100 [metric tons] x 0.02 [g of 239<\/sup>Pu\u00a0\/ g of fuel] . 106<\/sup>\u00a0[g\/metric ton]\n= 4\u00a0x 106<\/sup> grams of 239<\/sup>Pu<\/strong>\u00a0per the volume of the reactor core<\/p>\n

The atomic number density <\/a>of 239<\/sup>Pu\u00a0in the volume of the reactor core:<\/p>\n

N239<\/sub>\u00a0. V = m239<\/sub>\u00a0. NA<\/sub> \/ M239<\/sub>
\n= 4\u00a0x 106<\/sup> [g 239 \/ core] x 6.022 x 1023<\/sup>\u00a0[atoms\/mol] \/ 239 [g\/mol]\n= 10.1 x 1027<\/sup> atoms \/ core<\/strong>
\nThe microscopic fission cross-section of 235<\/sup>U (for thermal neutrons<\/a>):<\/p>\n

\u03c3f<\/sub>239<\/sup>\u00a0= 750 barns<\/strong><\/p>\n

The average recoverable energy per 239<\/sup>Pu\u00a0fission:<\/p>\n

Er<\/sub> = 207 MeV\/fission<\/strong><\/p>\n

\"Neutron<\/a><\/p>\n<\/div><\/div><\/div>\n

Neutron Flux – Uranium vs. MOX<\/h2>\n

Note that there is a difference between neutron fluxes in the uranium fueled core and the MOX fueled core. The average neutron flux in the first example, in which the neutron flux in a uranium-loaded reactor core<\/strong> was calculated, was 3.11 x 1013\u00a0<\/sup> neutrons.cm-2<\/sup>.s-1<\/sup><\/strong>. Compared to this value, the average neutron flux in 100% MOX fueled core<\/strong> is about 2.6 times lower<\/strong> (1.2 x 1013\u00a0<\/sup> neutrons.cm-2<\/sup>.s-1<\/sup><\/strong>), while the reaction rate remains almost the same. This fact is of importance in the reactor core design and the design of reactivity control. It is primarily caused by:<\/p>\n