{"id":13528,"date":"2017-01-03T12:25:44","date_gmt":"2017-01-03T12:25:44","guid":{"rendered":"http:\/\/sitepourvtc.com\/?page_id=13528"},"modified":"2022-03-06T12:35:23","modified_gmt":"2022-03-06T12:35:23","slug":"reaction-rate","status":"publish","type":"page","link":"https:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/reaction-rate\/","title":{"rendered":"Reaction Rate"},"content":{"rendered":"<div   id="8n9s8rpp6h"    class=\"su-quote su-quote-style-default\"><div   id="8n9s8rpp6h"    class=\"su-quote-inner su-u-clearfix su-u-trim\"><strong>The reaction rate<\/strong> is the number of interactions taking place in one cubic centimeter per one second. In a typical PWR reactor, the reaction rate is about 10<sup>20<\/sup> fissions\/second.<\/div><\/div>\n<p>Knowledge of the <a title=\"Neutron Flux \u2013 Neutron Intensity\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/neutron-flux-neutron-intensity\/\"><strong>neutron flux<\/strong> <\/a>(the <strong>total path length<\/strong> of all the neutrons in a cubic centimeter in a second) and the<a title=\"Macroscopic Cross-section\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/macroscopic-cross-section\/\"><strong> macroscopic cross sections<\/strong><\/a> (the probability of having an interaction <strong>per centimeter path length<\/strong>) allows us to compute the rate of interactions (e.g., rate of fission reactions). <strong>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<p><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/01\/Reaction-Rate-Neutron-Flux.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-13513\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/01\/Reaction-Rate-Neutron-Flux.png\" alt=\"Reaction Rate - Neutron Flux\" width=\"548\" height=\"77\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/01\/Reaction-Rate-Neutron-Flux.png 548w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/01\/Reaction-Rate-Neutron-Flux-300x42.png 300w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/01\/Reaction-Rate-Neutron-Flux-540x77.png 540w\" sizes=\"(max-width: 548px) 100vw, 548px\" \/><\/a><\/p>\n<p>where:<br \/>\n<strong>\u0424 &#8211; neutron flux (neutrons.cm<sup>-2<\/sup>.s<sup>-1<\/sup>)<\/strong><br \/>\n<strong>\u03c3 &#8211; microscopic cross section (cm<sup>2<\/sup>)<\/strong><br \/>\n<strong>N &#8211; atomic number density (atoms.cm<sup>-3<\/sup>)<\/strong><\/p>\n<p>The reaction rate for various types of interactions is found from the appropriate cross-section type:<\/p>\n<ul>\n<li><strong><strong>\u03a3<sub>t<\/sub> .\u00a0\u0424 &#8211; total reaction rate<\/strong><\/strong><\/li>\n<li><strong>\u03a3<sub>a<\/sub> .\u00a0\u0424 &#8211; absorption\u00a0reaction rate<\/strong>\n<ul>\n<li><strong><strong>\u03a3<sub>c<\/sub> .\u00a0\u0424 &#8211; radiative capture\u00a0reaction rate<\/strong><\/strong><\/li>\n<li><strong>\u03a3<sub>f<\/sub> .\u00a0\u0424 &#8211; fission\u00a0reaction rate<\/strong><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<div   id="8n9s8rpp6h"    class=\"collapsible-container\">\n<h2>Reaction Rate and Reactor Power Calculation<\/h2>\n<p>Multiplying the reaction rate per unit volume (RR = \u0424 . \u03a3) by the total volume of the core (V) gives us the<strong> total number of reactions<\/strong> occurring in the <a title=\"Reactor Core\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-reactor-core\/\">reactor core<\/a> per unit time. But we also know the amount of<a title=\"Energy Release from Fission\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/energy-release-from-fission\/\"> energy released per one fission reaction<\/a> to be about <strong>200 MeV\/fission<\/strong>. Now, it is possible to determine the <strong>rate of energy release<\/strong> (power) due to the fission reaction. It is given by following equation:<\/p>\n<p><strong>P = RR . E<sub>r<\/sub> . V = \u0424 . \u03a3<sub>f<\/sub> . E<sub>r<\/sub> . V = \u0424 . N<sub>U235<\/sub> . \u03c3<sub>f<\/sub><sup>235<\/sup> . E<sub>r<\/sub> . V<\/strong><\/p>\n<p>where:<br \/>\n<strong>P &#8211; reactor power (MeV.s<sup>-1<\/sup>)<\/strong><br \/>\n<strong>\u0424 &#8211; neutron flux (neutrons.cm<sup>-2<\/sup>.s<sup>-1<\/sup>)<\/strong><br \/>\n<strong>\u03c3 &#8211; microscopic cross section (cm<sup>2<\/sup>)<\/strong><br \/>\n<strong>N &#8211; atomic number density (atoms.cm<sup>-3<\/sup>)<\/strong><br \/>\n<strong>Er &#8211; the average recoverable energy per fission (MeV \/ fission)<\/strong><br \/>\n<strong>V &#8211; total volume of the core (m<sup>3<\/sup>)<\/strong><\/p>\n<h2>Example &#8211; Reaction Rate and Reactor Power<\/h2>\n<p>A typical <strong>thermal reactor<\/strong> contains about <strong>100 tons<\/strong> of<a title=\"Uranium\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-fuel\/uranium\/\"> uranium<\/a> with an average enrichment of <strong>2%<\/strong> (do not confuse it with the enrichment of the\u00a0<strong>fresh fuel<\/strong>). If the reactor power is 3000MW<sub>th<\/sub>, determine the <strong>reaction rate<\/strong> and the <strong>average core thermal flux<\/strong>.<\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>The amount of <a title=\"Fissile Material\" href=\"http:\/\/sitepourvtc.com\/glossary\/fissile-material\/\">fissile<\/a> <a title=\"Uranium 235\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-fuel\/uranium\/uranium-235\/\"><sup>235<\/sup>U<\/a> per the volume of the reactor core.<\/p>\n<p>m<sub>235<\/sub>\u00a0[g\/core] = 100 [metric tons] x 0.02 [g of <sup>235<\/sup>U\u00a0\/ g of U] . 10<sup>6<\/sup>\u00a0[g\/metric ton]\n= <strong>2 x 10<sup>6<\/sup> grams of <sup>235<\/sup>U<\/strong>\u00a0per the volume of the reactor core<\/p>\n<p>The <a title=\"Atomic Number Density\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/atomic-number-density\/\">atomic number density <\/a>of <sup>235<\/sup>U\u00a0in the volume of the reactor core:<\/p>\n<p>N<sub>235<\/sub> . V = m<sub>235<\/sub> . N<sub>A<\/sub> \/ M<sub>235<\/sub><br \/>\n= 2 x 10<sup>6<\/sup> [g 235 \/ core] x 6.022 x 10<sup>23<\/sup>\u00a0[atoms\/mol] \/ 235 [g\/mol]\n= <strong>5.13 x 10<sup>27<\/sup> atoms \/ core<\/strong><br \/>\nThe microscopic fission cross-section of <sup>235<\/sup>U (for <a title=\"Thermal Neutron\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/neutron\/thermal-neutron\/\">thermal neutrons<\/a>):<\/p>\n<p><strong>\u03c3<sub>f<\/sub><sup>235<\/sup> = 585 barns<\/strong><\/p>\n<p>The average recoverable energy per <sup>235<\/sup>U\u00a0fission:<\/p>\n<p><strong>E<sub>r<\/sub> = 200.7 MeV\/fission<\/strong><\/p>\n<p><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/01\/Neutron-Flux-Reaction-Rate-Thermal-Reactor.png\"><img loading=\"lazy\" class=\" size-full wp-image-13517 alignleft\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/01\/Neutron-Flux-Reaction-Rate-Thermal-Reactor.png\" alt=\"Neutron Flux - Reaction Rate - Thermal Reactor\" width=\"815\" height=\"427\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/01\/Neutron-Flux-Reaction-Rate-Thermal-Reactor.png 815w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/01\/Neutron-Flux-Reaction-Rate-Thermal-Reactor-300x157.png 300w\" sizes=\"(max-width: 815px) 100vw, 815px\" \/><\/a><\/p>\n<\/div>\n<div   id="8n9s8rpp6h"    class=\"su-divider su-divider-style-dotted\" style=\"margin:20px 0;border-width:2px;border-color:#999999\"><\/div>\n<div   id="8n9s8rpp6h"     class=\"lgc-column lgc-grid-parent lgc-grid-100 lgc-tablet-grid-100 lgc-mobile-grid-100 lgc-equal-heights \"><div   id="8n9s8rpp6h"     class=\"inside-grid-column\">\u00a0 <div   id="8n9s8rpp6h"    class=\"su-accordion su-u-trim\"><div   id="8n9s8rpp6h"    class=\"su-spoiler su-spoiler-style-default su-spoiler-icon-plus su-spoiler-closed\" data-scroll-offset=\"0\"><div   id="8n9s8rpp6h"    class=\"su-spoiler-title\" tabindex=\"0\" role=\"button\"><span class=\"su-spoiler-icon\"><\/span>References:<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\"><strong>Nuclear and Reactor Physics:<\/strong>\n<ol>\n<li>J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading,\u00a0MA (1983).<\/li>\n<li>J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.<\/li>\n<li>W. M. Stacey, Nuclear Reactor Physics, John Wiley &amp; Sons, 2001, ISBN: 0- 471-39127-1.<\/li>\n<li>Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering,\u00a0Springer; 4th edition, 1994, ISBN:\u00a0978-0412985317<\/li>\n<li>W.S.C. Williams. Nuclear and Particle Physics.\u00a0Clarendon Press; 1 edition, 1991, ISBN:\u00a0978-0198520467<\/li>\n<li>G.R.Keepin. Physics of Nuclear Kinetics.\u00a0Addison-Wesley Pub. Co; 1st edition, 1965<\/li>\n<li>Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.<\/li>\n<li>U.S. Department of Energy, Nuclear Physics and Reactor Theory.\u00a0DOE Fundamentals Handbook,\u00a0Volume 1 and 2.\u00a0January\u00a01993.<\/li>\n<\/ol>\n<p><strong><\/strong><strong>Advanced Reactor Physics:<\/strong><\/p>\n<ol>\n<li>K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.<\/li>\n<li>K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.<\/li>\n<li><span style=\"font-weight: 400;\">D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2.\u00a0<\/span><\/li>\n<li>E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.<\/li>\n<\/ol>\n<\/div><\/div><\/div><div   id="8n9s8rpp6h"    class=\"su-divider su-divider-style-dotted\" style=\"margin:15px 0;border-width:2px;border-color:#999999\"><\/div><\/div><\/div><div   id="8n9s8rpp6h"    class=\"su-divider su-divider-style-default\" style=\"margin:15px 0;border-width:2px;border-color:#999999\"><\/div><div   id="8n9s8rpp6h"     class=\"lgc-column lgc-grid-parent lgc-grid-33 lgc-tablet-grid-33 lgc-mobile-grid-100 lgc-equal-heights \"><div   id="8n9s8rpp6h"     class=\"inside-grid-column\">\n<h2>See previous:<\/h2>\n<a href=\"\" class=\"su-button su-button-style-flat\" style=\"color:#606060;background-color:#ffffff;border-color:#cccccc;border-radius:10px;-moz-border-radius:10px;-webkit-border-radius:10px\" target=\"_self\"><span style=\"color:#606060;padding:7px 20px;font-size:16px;line-height:24px;border-color:#ffffff;border-radius:10px;-moz-border-radius:10px;-webkit-border-radius:10px;text-shadow:0px 0px 0px #000000;-moz-text-shadow:0px 0px 0px #000000;-webkit-text-shadow:0px 0px 0px #000000\"><i class=\"sui sui-link\" style=\"font-size:16px;color:#5d5d5d\"><\/i> <\/span><\/a><\/div><\/div><div   id="8n9s8rpp6h"     class=\"lgc-column lgc-grid-parent lgc-grid-33 lgc-tablet-grid-33 lgc-mobile-grid-100 lgc-equal-heights \"><div   id="8n9s8rpp6h"     class=\"inside-grid-column\">\n<h2>See above:<\/h2>\n<p>Neutron Nuclear Reactions<a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/\" class=\"su-button su-button-style-flat\" style=\"color:#606060;background-color:#ffffff;border-color:#cccccc;border-radius:10px;-moz-border-radius:10px;-webkit-border-radius:10px\" target=\"_self\"><span style=\"color:#606060;padding:7px 20px;font-size:16px;line-height:24px;border-color:#ffffff;border-radius:10px;-moz-border-radius:10px;-webkit-border-radius:10px;text-shadow:0px 0px 0px #000000;-moz-text-shadow:0px 0px 0px #000000;-webkit-text-shadow:0px 0px 0px #000000\"><i class=\"sui sui-link\" style=\"font-size:16px;color:#5d5d5d\"><\/i> <\/span><\/a><\/p><\/div><\/div><div   id="8n9s8rpp6h"     class=\"lgc-column lgc-grid-parent lgc-grid-33 lgc-tablet-grid-33 lgc-mobile-grid-100 lgc-equal-heights \"><div   id="8n9s8rpp6h"     class=\"inside-grid-column\">\n<h2>See next:<\/h2>\n<a href=\"\" class=\"su-button su-button-style-flat\" style=\"color:#606060;background-color:#ffffff;border-color:#cccccc;border-radius:10px;-moz-border-radius:10px;-webkit-border-radius:10px\" target=\"_self\"><span style=\"color:#606060;padding:7px 20px;font-size:16px;line-height:24px;border-color:#ffffff;border-radius:10px;-moz-border-radius:10px;-webkit-border-radius:10px;text-shadow:0px 0px 0px #000000;-moz-text-shadow:0px 0px 0px #000000;-webkit-text-shadow:0px 0px 0px #000000\"><i class=\"sui sui-link\" style=\"font-size:16px;color:#5d5d5d\"><\/i> <\/span><\/a><\/div><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Knowledge of the neutron flux (the total path length of all the neutrons in a cubic centimeter in a second) and the macroscopic cross sections (the probability of having an interaction per centimeter path length) allows us to compute the rate of interactions (e.g., rate of fission reactions). The reaction rate (the number of interactions &#8230; <a title=\"Reaction Rate\" class=\"read-more\" href=\"https:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/reaction-rate\/\" aria-label=\"More on Reaction Rate\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":12656,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"generate_page_header":""},"_links":{"self":[{"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/pages\/13528"}],"collection":[{"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/comments?post=13528"}],"version-history":[{"count":3,"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/pages\/13528\/revisions"}],"predecessor-version":[{"id":31585,"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/pages\/13528\/revisions\/31585"}],"up":[{"embeddable":true,"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/pages\/12656"}],"wp:attachment":[{"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/media?parent=13528"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}