{"id":13658,"date":"2017-02-07T08:48:35","date_gmt":"2017-02-07T08:48:35","guid":{"rendered":"http:\/\/sitepourvtc.com\/?page_id=13658"},"modified":"2022-10-21T07:06:33","modified_gmt":"2022-10-21T07:06:33","slug":"four-factor-formula-infinite-multiplication-factor","status":"publish","type":"page","link":"https:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/four-factor-formula-infinite-multiplication-factor\/","title":{"rendered":"Four-Factor Formula &#8211; Infinite Multiplication Factor"},"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\">The <strong>infinite multiplication factor<\/strong> is the ratio of the <strong>neutrons produced by fission<\/strong> in one <a title=\"Neutron Generation \u2013 Neutron Population\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/neutron-generation-neutron-population\/\">neutron generation<\/a> to the number of <strong>neutrons lost through absorption<\/strong> in the preceding neutron generation. The infinite multiplication factor (k<sub>\u221e<\/sub>) may be expressed mathematically in terms of these factors by the following equation, usually known as the <strong>four-factor formula<\/strong>:<\/p>\n<p style=\"text-align: center;\"><strong>k<sub>\u221e<\/sub>\u00a0= \u03b7.\u03b5.p.f<\/strong><\/p>\n<\/div><\/div>\n<p>In this section, <strong>the infinite multiplication factor<\/strong>, which describes all the possible events in the life of a neutron and effectively describes the state of an infinite multiplying system, will be defined.<\/p>\n<p>The necessary condition for a <strong>stable, self-sustained fission chain reaction<\/strong> in a multiplying system (in a <a title=\"Nuclear Reactor\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-reactor\/\">nuclear reactor<\/a>) is that <strong>exactly every\u00a0<a title=\"Nuclear Fission\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/\">fission <\/a>initiates another fission<\/strong>. The minimum condition is for each nucleus undergoing fission to produce, on average, at least one neutron that causes fission of another nucleus. Also, the number of fissions occurring per unit time (<a title=\"Reaction Rate\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/reaction-rate\/\">the reaction rate<\/a>) within the system must be constant.<\/p>\n<p>This condition can be expressed conveniently in terms of <strong>the multiplication factor<\/strong>. The infinite multiplication factor is the ratio of the <strong>neutrons produced by fission<\/strong> in one <a title=\"Neutron Generation \u2013 Neutron Population\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/neutron-generation-neutron-population\/\">neutron generation<\/a> to the number of <strong>neutrons lost through absorption<\/strong> in the preceding neutron generation. This can be expressed mathematically, as shown below.<\/p>\n<p><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/02\/Multiplication-Factor.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-13628\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/02\/Multiplication-Factor.png\" alt=\"Multiplication Factor\" width=\"485\" height=\"71\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/02\/Multiplication-Factor.png 485w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/02\/Multiplication-Factor-300x44.png 300w\" sizes=\"(max-width: 485px) 100vw, 485px\" \/><\/a><\/p>\n<p>The<strong> infinite multiplication factor<\/strong> in a multiplying system measures the change in the fission <a title=\"Neutron Generation \u2013 Neutron Population\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/neutron-generation-neutron-population\/\">neutron population<\/a> from one neutron generation to the subsequent\u00a0generation.<\/p>\n<ul>\n<li><strong>k<sub>\u221e<\/sub>\u00a0&lt; 1<\/strong>. Suppose the multiplication factor for a multiplying system is <strong>less than 1.0<\/strong>. In that case, the <strong>number of neutrons decreases<\/strong> in time (with the <a title=\"Mean Generation Time with Delayed Neutrons\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/delayed-neutrons\/mean-generation-time-with-delayed-neutrons\/\">mean generation time<\/a>), and the chain reaction will never be self-sustaining. This condition is known as <strong>the subcritical state<\/strong>.<\/li>\n<\/ul>\n<ul>\n<li><strong>k<sub>\u221e<\/sub> = 1<\/strong>. If the multiplication factor for a multiplying system is <strong>equal to 1.0<\/strong>, then there is <strong>no change in neutron population<\/strong> in time, and the chain reaction will be<strong> self-sustaining<\/strong>. This condition is known as <strong>the critical state<\/strong>.<\/li>\n<\/ul>\n<ul>\n<li><strong>k<sub>\u221e<\/sub>\u00a0&gt; 1<\/strong>. If the multiplication factor for a multiplying system is <strong>greater than 1.0<\/strong>, then the multiplying system produces <strong>more neutrons<\/strong>\u00a0than are needed to be self-sustaining. The number of neutrons is exponentially increasing in time (with the mean generation time). This condition is known as <strong>the supercritical state<\/strong>.<\/li>\n<\/ul>\n<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>From the multiplication factor to the reactor control<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\"><strong><span style=\"font-weight: 400;\">The simplest equation governing the neutron kinetics of the system with delayed neutrons is<\/span><\/strong>\u00a0<strong>the\u00a0point kinetics equation<\/strong>. This equation\u00a0states that the time change of the neutron population is equal\u00a0to the <strong>excess of neutron production<\/strong> (by <a title=\"Fast Fission Factor\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/fast-fission-factor\/\">fission<\/a>) <strong>minus neutron loss<\/strong> by absorption <strong>in one <a title=\"Mean Generation Time with Delayed Neutrons\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/delayed-neutrons\/mean-generation-time-with-delayed-neutrons\/\">mean generation time with delayed neutrons<\/a> (l<sub>d<\/sub>)<\/strong>. The role of <strong>l<sub>d<\/sub><\/strong>\u00a0and <strong>k<sub>eff\u00a0<\/sub><\/strong>is evident. Larger <strong>k<sub>eff<\/sub><\/strong> give a larger response and l<strong><span style=\"font-weight: 400;\">onger lifetimes give simply slower responses of multiplying systems.<\/span><\/strong><\/p>\n<p>If there are neutrons in the system at t=0, that is, if n(0) &gt; 0, the solution of this equation gives\u00a0<strong><span style=\"font-weight: 400;\">the simplest point kinetics equation with delayed neutrons (similarly to the case <a title=\"Infinite Multiplying System Without Source and Delayed Neutrons\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/fission\/prompt-neutrons\/infinite-multiplying-system-without-source-and-delayed-neutrons\/\">without delayed neutrons<\/a>):<\/span><\/strong><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/11\/point-kinetics-equation-with-delayed-neutrons.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-12892\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2015\/11\/point-kinetics-equation-with-delayed-neutrons.png\" alt=\"point kinetics equation with delayed neutrons\" width=\"282\" height=\"152\" \/><\/a><\/p>\n<p>Let us consider that <strong>the mean generation time with delayed neutrons is ~0.085,<\/strong>\u00a0and k (k<sub>eff<\/sub> &#8211; neutron multiplication factor) will be increased <strong>by only 0.01%<\/strong> (<strong>i.e., 10pcm or ~1.5 cents<\/strong>), k<sub>\u221e<\/sub>=1.0000 will increase to k<sub>\u221e<\/sub>=1.0001.<\/p>\n<p>It must be noted such reactivity insertion (10pcm)<strong> is very small<\/strong> in the case of LWRs. The reactivity insertions <strong>of the order of one pcm<\/strong> are for LWRs <strong>practically unrealizable<\/strong>. In this case, the reactor period will be:<\/p>\n<p style=\"text-align: center;\"><strong>T = l<sub>d<\/sub> \/ (k<sub>\u221e<\/sub>-1) = 0.085 \/ (1.0001-1) = 850s<\/strong><\/p>\n<p>This is a very long period. In ~14 minutes, the reactor&#8217;s <a title=\"Neutron Flux Density \u2013 Neutron Intensity\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/neutron-flux-neutron-intensity\/\">neutron flux<\/a> (and reactor power) would increase by a factor of e = 2.718. This is a completely different dimension of the response on reactivity insertion compared to the case without delayed neutrons, where the reactor period was 1 second.<\/p>\n<p>Reactors with such kinetics would be quite <strong>easy to control<\/strong>. From this point of view, it may seem that reactor control will be quite a boring affair. It will not! The presence of delayed neutrons entails many specific phenomena that will be described in later chapters.<\/p>\n<\/div><\/div><\/div>\n<div   id="8n9s8rpp6h"    class=\"collapsible-container\">\n<h2>Four-Factor Formula<\/h2>\n<p>But <strong>the infinite multiplication factor<\/strong> can also be defined in terms of the most important <strong>neutron-physical processes<\/strong> in the<a title=\"Nuclear Reactor\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-reactor\/\"> nuclear reactor<\/a>. For simplicity, we will first consider a multiplying system that is <strong>infinitely large<\/strong>\u00a0and therefore has <strong>no neutron leakage<\/strong>. In the infinite system. <strong>Four-factors<\/strong> are completely independent of the size and shape of the reactor that describes\u00a0the <strong>inherent multiplication ability<\/strong> of the <a title=\"Nuclear Fuel\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-fuel\/\">fuel<\/a> and <a title=\"Neutron Moderator\" href=\"http:\/\/sitepourvtc.com\/neutron-moderator\/\">moderator<\/a> materials without regard to leakage:<\/p>\n<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>Fast Fission Factor <\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\"><strong>The fast fission process<\/strong> is in the multiplication factor characterized by the <strong>fast fission factor, \u03b5<\/strong>, which increases the fast neutron population in one neutron generation. The fast fission factor is the ratio of the fast neutrons produced by fissions at all energies to the number of fast neutrons produced in thermal fission.<\/div><\/div><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>Resonance Escape Probability<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\"><strong>The resonance escape probability<\/strong>, symbolized by p, is the probability that a neutron will be slowed to thermal energy and will escape <a title=\"What is Nuclear Resonance \u2013 Compound Nucleus\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/compound-nucleus-reactions\/what-is-nuclear-resonance-compound-nucleus\/\">resonance capture<\/a>. This probability is defined as the ratio of the number of neutrons that reach thermal energies to the number of fast neutrons that slow down.<\/div><\/div><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>Thermal Utilization Factor<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\"><strong>The thermal utilization factor, f<\/strong>, is the fraction of the <a title=\"Thermal Neutron\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/neutron\/thermal-neutron\/\">thermal neutrons<\/a> that are absorbed in the<a title=\"Nuclear Fuel\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-fuel\/\"> nuclear fuel<\/a>, in all isotopes of the nuclear fuel. It describes how effectively (how well they are utilized) thermal neutrons are absorbed in the fuel. 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 <a title=\"Reactor Core\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-reactor-core\/\">core<\/a>.<\/div><\/div><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>Reproduction Factor<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\">The number of neutrons created in the new generation is determined by<strong> the neutron reproduction factor<\/strong>. <strong>The reproduction factor, \u03b7<\/strong>, is defined as the ratio of the number of fast neutrons produced by thermal fission to the number of <a title=\"Thermal Neutron\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/atomic-nuclear-physics\/fundamental-particles\/neutron\/thermal-neutron\/\">thermal neutrons<\/a> absorbed in the fuel.<\/div><\/div><\/div>\n<p>See also: <a title=\"Fast Fission Factor\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/fast-fission-factor\/\">Fast Fission Factor<\/a><\/p>\n<p>See also: <a title=\"Resonance Escape Probability\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/resonance-escape-probability\/\">Resonance Escape Probability<\/a><\/p>\n<p>See also: <a title=\"Thermal Utilization Factor\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/thermal-utilization-factor\/\">Thermal Utilization Factor<\/a><\/p>\n<p>See also: <a title=\"Reproduction Factor\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/reproduction-factor\/\">Reproduction Factor<\/a><\/p>\n<p>The infinite multiplication factor (k<sub>\u221e<\/sub>) may be expressed mathematically in terms of these factors by the following equation, usually known as the <strong>four-factor formula<\/strong>:<\/p>\n<p style=\"text-align: center;\"><strong>k<sub>\u221e<\/sub>\u00a0= \u03b7.\u03b5.p.f<\/strong><\/p>\n<p>In reactor physics, <strong>k<sub>\u221e<\/sub><\/strong>\u00a0or its finite form <strong>k<sub>eff<\/sub><\/strong>\u00a0is the most significant parameter with regard to reactor control. At any specific power level or condition of the reactor, <strong>k<sub>eff<\/sub><\/strong> is kept as near to the value of <strong>1.0<\/strong> as possible. At this point in the operation, the <strong>neutron balance<\/strong> is kept to exactly one neutron completing the life cycle for each original neutron absorbed in the fuel.<\/p>\n<h2>From infinite to an effective multiplication factor<\/h2>\n<p>The infinite multiplication factor is derived based on the assumption that <strong>no neutrons leak out of the reactor<\/strong> (i.e., a reactor is infinitely large). But in reality, each nuclear reactor is finite, and neutrons can leak out of the <a title=\"Reactor Core\" href=\"http:\/\/sitepourvtc.com\/nuclear-power-plant\/nuclear-reactor-core\/\">reactor core<\/a>. The multiplication factor that takes <strong>neutron leakage<\/strong> into account is the <a title=\"Six Factor Formula \u2013 Effective Multiplication Factor\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/six-factor-formula-effective-multiplication-factor\/\"><strong>effective multiplication factor<\/strong> &#8211; <strong>k<sub>eff<\/sub><\/strong><\/a>, which is the ratio of the <strong>neutrons produced by fission<\/strong> in one neutron generation to the number of <strong>neutrons lost through absorption and leakage<\/strong> in the preceding<a title=\"Neutron Generation \u2013 Neutron Population\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/neutron-generation-neutron-population\/\"> neutron generation<\/a>. The effective multiplication factor (<strong>k<sub>eff<\/sub><\/strong>) may be expressed mathematically in terms of the infinite multiplication factor (k<sub>\u221e<\/sub>) and two additional factors which account for <strong>neutron leakage<\/strong> during neutron thermalization (<strong>fast non-leakage probability<\/strong>) and neutron leakage during neutron diffusion (<strong>thermal non-leakage probability<\/strong>) by following equation, usually known as the <strong>six-factor formula<\/strong>:<\/p>\n<p style=\"text-align: center;\"><strong>k<sub>eff<\/sub> = k<sub>\u221e<\/sub> . P<sub>f<\/sub> . P<sub>t<\/sub><\/strong><\/p>\n<p>&nbsp;<\/p>\n<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>Six Factor Formula - Fast Reactors<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\">The method of calculations of multiplication factors has been developed <strong>in the early years<\/strong> of nuclear energy and is only applicable to <strong>thermal reactors<\/strong>, where the bulk of fission reactions occurs at thermal energies. This method puts into context all the processes associated with the thermal reactors (e.g.,, neutron thermalization, neutron diffusion, or fast fission) because the most important neutron-physical processes occur in energy <strong>regions that can be clearly separated from each other<\/strong>. In short, the calculation of the multiplication factor gives a good insight into the processes that occur in each thermal multiplying system.<\/p>\n<p>For<a title=\"Fast Neutron Reactor\" href=\"http:\/\/sitepourvtc.com\/fast-neutron-reactor\/\"><strong> fast reactors<\/strong><\/a>, in which the fission is caused by neutrons with a very broad energy distribution, such an analysis is inappropriate. The <a title=\"Neutron Flux Spectra\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-engineering-fundamentals\/neutron-nuclear-reactions\/neutron-flux-spectra\/\">neutron flux<\/a> in fast reactors has to be divided into<strong> many energy groups<\/strong>. Moreover, in fast reactors, neutron thermalization is an undesirable process, and therefore the four-factor formula does not make any sense. The resonance escape probability is insignificant because very few neutrons exist at energies where resonance absorption is significant. The thermal non-leakage probability does not exist because the reactor is designed to avoid the thermalization of neutrons.<\/p>\n<\/div><\/div><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>Fast vs. Thermal Flux Spectrum<\/div><div   id="8n9s8rpp6h"    class=\"su-spoiler-content su-u-clearfix su-u-trim\"><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/01\/thermal-vs-fast-reactor-neutron-spectrum-min.png\"><img loading=\"lazy\" class=\"size-large wp-image-13563\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/01\/thermal-vs-fast-reactor-neutron-spectrum-min-1024x1024.png\" alt=\"thermal vs. fast reactor neutron spectrum\" width=\"669\" height=\"669\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/01\/thermal-vs-fast-reactor-neutron-spectrum-min-1024x1024.png 1024w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/01\/thermal-vs-fast-reactor-neutron-spectrum-min-150x150.png 150w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/01\/thermal-vs-fast-reactor-neutron-spectrum-min-300x300.png 300w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/01\/thermal-vs-fast-reactor-neutron-spectrum-min-66x66.png 66w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/01\/thermal-vs-fast-reactor-neutron-spectrum-min.png 1280w\" sizes=\"(max-width: 669px) 100vw, 669px\" \/><\/a>The spectrum of neutron energies produced by fission vary significantly with certain reactor design. thermal vs. fast reactor neutron spectrum<\/div><\/div><\/div>\n<p><a href=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/02\/Six-Factor-Formula-Four-Factor-Formula.png\"><img loading=\"lazy\" class=\"aligncenter size-full wp-image-13834\" src=\"http:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/02\/Six-Factor-Formula-Four-Factor-Formula.png\" alt=\"Six Factor Formula - Four Factor Formula\" width=\"839\" height=\"790\" srcset=\"https:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/02\/Six-Factor-Formula-Four-Factor-Formula.png 839w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/02\/Six-Factor-Formula-Four-Factor-Formula-300x282.png 300w, https:\/\/sitepourvtc.com\/wp-content\/uploads\/2016\/02\/Six-Factor-Formula-Four-Factor-Formula-52x50.png 52w\" sizes=\"(max-width: 839px) 100vw, 839px\" \/><\/a><\/p>\n<h2>Operational factors that affect the multiplication in PWRs.<\/h2>\n<p>Detailed knowledge of all possible operational factors that may affect the multiplication factor of the system is of importance in <strong>reactor control<\/strong>. It was stated the <strong>k<sub>eff<\/sub> <\/strong>is during reactor operation kept as near to the value of <strong>1.0 as possible<\/strong>. Many factors influence <b>the criticality of the reactor<\/b>. For illustration, in an extreme case, the presence of humans (due to the water, carbon, which are good <a title=\"Neutron Moderator\" href=\"http:\/\/sitepourvtc.com\/neutron-moderator\/\">neutron moderators<\/a>) near fresh uranium fuel assembly influences the multiplication properties of the assembly. If any operational factor changes one of the contributing factors to <strong>k<sub>eff<\/sub><\/strong>\u00a0(<strong>k<sub>eff<\/sub> = \u03b7.\u03b5.p.f.P<sub>f<\/sub>.P<sub>t<\/sub><\/strong>), the ratio of 1.0 is not maintained. This change in <strong>k<sub>eff<\/sub><\/strong>\u00a0makes the reactor either <strong>subcritical<\/strong> or <strong>supercritical<\/strong>. Some examples of these operational changes in PWRs are below and are described in a separate article in detail.<\/p>\n<ul>\n<li><strong>change in the control rods position<\/strong><\/li>\n<li><strong>change in the boron concentration<\/strong><\/li>\n<li><strong>change in the moderator temperature<\/strong><\/li>\n<li><strong>change in the fuel temperature<\/strong><\/li>\n<li><strong>change in the pressure<\/strong><\/li>\n<li><strong>change in the flow rate<\/strong><\/li>\n<li><strong>presence of boiling of the coolant<\/strong><\/li>\n<li><strong>presence of burnable absorbers<\/strong><\/li>\n<li><strong>fuel burnup<\/strong><\/li>\n<\/ul>\n<p>See also: <a title=\"Operational factors that affect the multiplication in PWRs\" href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/operational-factors\/\">Operational changes that affect the multiplication in PWRs.<\/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>Chain Reaction<a href=\"http:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/\" 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>In this section, the infinite multiplication factor, which describes all the possible events in the life of a neutron and effectively describes the state of an infinite multiplying system, will be defined. The necessary condition for a stable, self-sustained fission chain reaction in a multiplying system (in a nuclear reactor) is that exactly every\u00a0fission initiates &#8230; <a title=\"Four-Factor Formula &#8211; Infinite Multiplication Factor\" class=\"read-more\" href=\"https:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/nuclear-fission-chain-reaction\/four-factor-formula-infinite-multiplication-factor\/\" aria-label=\"More on Four-Factor Formula &#8211; Infinite Multiplication Factor\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":13612,"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\/13658"}],"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=13658"}],"version-history":[{"count":5,"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/pages\/13658\/revisions"}],"predecessor-version":[{"id":32848,"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/pages\/13658\/revisions\/32848"}],"up":[{"embeddable":true,"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/pages\/13612"}],"wp:attachment":[{"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/media?parent=13658"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}