Four factor formula

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TEENA THANKACHENREG no: 630-14126012


NUCLEAR FISSIONChief method of producing energy.Tremendous amount of energy is released.When heavy nuclei are bombarded with protons, deuterons, neutrons, particles etc , then the nucleus is caused to breakdown into two roughily equal parts , known as fission fragments. This process is called NUCLEAR FISSION.Frisch and Meitner in 1939 used the word FISSION .

NUCLEAR CHAIN REACTIONWhen a neutron produces fission in uranium nucleus, besides the fission fragments a few fast neutrons are also emitted. If one or more of the emitted neutrons are used to fission of other nuclei, further neutrons are produced and the process is repeated. The reaction thus becomes self-sustained and is known as CHAIN REACTION.


NUCLEAR CHAIN REACTIONThe reaction is controlled in such a way that only one of the neutrons emitted in a fission causes another fission, then the fission rate remains constant and the energy released steadily. Such a reaction is called CONTROLLED CHAIN REACTION.It is used in Nuclear Reactors.


NEUTRON LIFE CYCLEIn thermal reactors,neutrons that cause fission are born at a much higher energy level than requiredTo make fission more probable,these neutrons must be slowed to thermal energy PWRs use water as a moderatorWhen moderating fast neutrons,gains and losses occur

This process is referred to as NEUTRON LIFE CYCLE .Explains factors involved in controlling nuclear fission rateProper management of the neutron life cycle makes control of a nuclear reactor possible

Some of the fast neutrons born by fission in one generation will cause fission in the next generationButFission neutrons travel through a series of events as they slow to thermal energies, leak, or are absorbed in the reactor Referred to as the neutron life cycle

Simplified neutron life cycle: All neutrons are born as fast neutronsSome fast neutrons are absorbed by fuel and cause fast fissionSome fast neutrons leak out of reactor coreSome fast neutrons undergo resonance capture while slowing downAll remaining fast neutrons become thermalized

Some thermal neutrons leak out of coreSome thermal neutrons absorbed by non-fuel materialSome thermal neutrons absorbed by fuel and not cause fissionRemaining thermal neutrons absorbed by fuel and cause thermal fission

Neutron production from fission in one generationNeutron absorption in the preceding generation

EFFECTIVE MULTIPLICATION FACTOR KeffDescribes neutron life cycle in a real, finite reactorA reactor of finite size will have neutrons leak out of itDefined as ratio of neutrons produced by fission in one generation to number of neutrons lost through absorption and leakage in preceding generation

Like K, by its value, tells whether a new generation of neutrons is larger, smaller, or same size as preceding generation Also known as six-factor formula

INFINITE VS. EFFECTIVE MULTIPLICATION FACTORIf leakage is small enough to be neglected, multiplication factor depends only on balance between production and absorption called Infinite multiplication factor Also called four-factor formula , considers factors shown below:

With leakage included,considers six factors

FOUR-FACTOR FORMULAAlso known as Infinite Multiplication Factor Used to consider a reactor of infinitely large size where no neutron leakage can occurDefined at ratio of neutrons produced by fission in one generation to number of neutrons lost through absorption in preceding generation

EFFECTIVE MULTIPLICATION FACTOR (KEFF) & CRITICALITYWhen value of keff is 1, a self-sustaining chain reaction of fissions is occurringNeutron population is neither increasing nor decreasingCalled critical or critical reactor keff = 1When neutron production is greater than the losses due to absorption and leakageReactor is supercriticalkeff > 1Neutron flux is increasing each generation

EFFECTIVE MULTIPLICATION FACTOR (KEFF) & CRITICALITYWhen neutron production is less than losses due to absorption and leakageReactor is subcriticalKeff < 1Neutron flux is decreasing each generationWhen keff is not equal to exactly 1, neutron flux and therefore reactor power will be changing

INFINITE MULTIPLICATION FACTOR Four factors independent of size and shape of reactor and do not consider any neutron leakage from the reactor.

Where :

= fast fission factor

= resonance escape probability

= reproduction factor

= thermal utilization factor


= No of fast neutrons produced by all fissions No of fast neutrons produced by thermal fissionsFirst event neutrons incur after birthCaused by neutrons that are in fast energy rangeResults in a net increase in fast neutron population

Neutrons must pass close to a fuel nucleus while still fastValue affected by fuel concentration and physical arrangement proximity to moderatorEssentially 1.00 for a homogenous reactor, fuel atoms surrounded by moderator atoms (rapid moderation)

Cross-section for fast fission in uranium-235 or uranium-238 is smallStill an appreciable number of fast neutrons cause fission in uranium-235, uranium-238, and plutomium-239A large fraction of fast fissions occur with uranium-235 because of its wider fission energy spectrum

In a heterogeneous reactor (PWR/BWR), fuel atoms packed closely together in fuel pellets within fuel rods and assemblies Neutrons emitted from fission of one fuel atom have a good chance of passing near another fuel atom before slowing downResults in some fast fissionFor PWRs, 1.02 is a good value for , with a range of 1.02 to 1.05


= No: of neutrons that reach thermal energyNo: of fast neutrons that starts to slow downAfter fast fissions occur, neutrons continue to diffuse throughout reactorCollide with nuclei of fuel, non-fuel material, and moderatorLose energy in each collision and slow down

All nuclei within reactor core have some probability of absorbing neutrons Microscopic cross-section for absorption (a) for each materiala is not a constant value, dependent on energy level of incident neutronAbsorption cross-sections increase as neutron energy level decreases

THERMAL UTILIZATION FACTOR (f)f=No:of thermal neutrons absorbed in the fuel No: of thermal neutrons absorbed in all reactor materialsAfter thermal non-leakage, thermalized neutrons still dispersed throughout the core where they are subject to absorption by either fuel or non-fuel material

Thermal utilization factor describes how effectively thermal neutrons are being absorbed by fuel or underutilized by non-fuel materialsThermal utilization factor is always less than oneNot all thermal neutrons are absorbed in fuelThese neutrons are lost to the fission processA value range for thermal utilization factor is 0.70-0.80

REPRODUCTION FACTOR ()= No: of fast neutrons neutrons produced by thermal fission

No: of thermal neutrons absorbed in the fuel

Most neutrons absorbed in fuel cause fission, but some do notReproduction factor represents net gain in neutron populationValue range of 1.65-2.0