Time Dep Reactor

7/24/2019 Time Dep Reactor

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INDIAN INSTITUTE OF TECHNOLOGY ROORKEE

IPH 305: Reactor PhysicsIPH 305: Reactor Physics

Ajay Y. Deo

Department of Physics

Room No. 305, Ph.: 5566

Time Dependent ReactorTime

Dependent Reactor


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ObjectiveObjective

To consider the properties when the reactor it isTo consider the properties when the reactor it is NOTNOTcritical.critical.

1. It was assumed in all earlier discussions that the rector wasIt was assumed in all earlier discussions that the rector wascritical and operating at a constant power.critical and operating at a constant power.

2.2. However, this is NOT always true. For instance, at the startupHowever, this is NOT always true. For instance, at the startupthe reactor must be supercritical to reach the desired power. hile itthe reactor must be supercritical to reach the desired power. hile it

must be subcritical to shut it down or to reduce the power. The studymust be subcritical to shut it down or to reduce the power. The studyo! reactor in a non"critical state is calledo! reactor in a non"critical state is called #eactor $inetics#eactor $inetics..

%.%. The degree o! criticality is regulated by control rods. &lso, theThe degree o! criticality is regulated by control rods. &lso, theparameters entering into the value o!parameters entering into the value o! kkare temperature dependent.are temperature dependent.

'.'. Two !ission products are produced in each !ission. (ertain o!Two !ission products are produced in each !ission. (ertain o!these nuclei ) especiallythese nuclei ) especially 1%*1%*+e +e 1'-1'-m have large absorption cross"m have large absorption cross"

section. Their presence in the reactor has considerable e!!ect on thesection. Their presence in the reactor has considerable e!!ect on thevalue o! /. uchvalue o! /. uch fission product poisonsfission product poisonswill also be discussed.will also be discussed.


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Reactor KineticsReactor Kinetics

0ost o! the neutrons are emitted at the instance o! !ission prompt neutronsprompt neutrons.

However, a small !raction o! neutrons appear long a!ter the !ission eventdelayed neutronsdelayed neutrons. The time behavior o! a reactor depends on the variousproperties o! these neutrons.

Prompt Neutron LifetimePrompt Neutron Lifetime

The average time between the emission o! the prompt neutrons and theirabsorption in the reactor is called theprompt neutron lifetimeand is denoted

bylp

.

Consider an infinite thermal reactor:

It is e3perimentally observed that the time re4uired to slow down a !astneutron to thermal energies is much smaller than the time it spends as athermal neutron be!ore it is !inally absorbed.

The average li!etime o! a thermal neutron in an in!inite system is called mean

diffusion timeand is denoted by td. This implies


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Reactor KineticsReactor KineticsCalculation of t

d

&verage distance traveled by a thermal neutron o! energy 5 be!ore it is

absorbed is

There!ore, it survives !or the time

The mean di!!usion time is then the average o! the above 4uantity

tEcan also be written as

&t thermal energies,

There!ore,

From thediscussion on thethermal neutrondi!!usion.

6sing ga7 1tt((EE) is constant, independent of) is constant, independent of EE


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Reactor KineticsReactor Kinetics

Further,

0ean di!!usion time !or the moderator0ean di!!usion time !or the moderator

Note:

The above analysis is valid ONLY for thermal reactors. The valuesThe above analysis is valid ONLY for thermal reactors. The values

of diffusion times are on the order of 10of diffusion times are on the order of 10-2-2to 10to 10-4-4sec. While for a fastsec. While for a fast

reactor these values are on the order of 10reactor these values are on the order of 10 -!-!sec.sec.


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Reactor KineticsReactor KineticsReactor with NO Delayed Neutrons

The delayed neutrons play an important role in reactor /inetics. These neutrons

!orm only 18 part o! the total neutrons emitted in the thermal !ission o!2%*

6.To understand this e3traordinary !act, let us !irst consider a in!inite thermalreactor in the absence o! delayed neutrons.

The absorption o! a !ission neutron lead to generation o! !ission neutrons.

The mean "eneration timeThe mean "eneration timeis de!ined as the time between the birth o! a

neutron and subse4uent absorption ) inducing !ission.

In the typical case !or k7 1 and with no delayed neutrons, it is e4ual to lp.

The absorption o! a neutron in one generation leads to the absorption o!

neutrons in the ne3t generation a!ter lp

sec.

Thus, i!NFt is the number o! !issions occurring per cm%9sec at time t, then

the !ission rate a!ter lp

sec is


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Solution

Fission rate at t7 :

#eactor $eriod#eactor $eriod

For an infinite, critical thermal reactor with homogeneous mixture of water

and 235U, the prompt neutron lifetime is 10-4sec. If such a reactor is critical

upto time t= 0, and then if is increased from 1.000 to 1.001. Compute

the response of the reactor to this change in .

Answer

This implies that the !lu3 and hence the power will increase as e10t, where tis in sec.


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In the e3ample discussed, the reactor period is :.1 sec. This implies that

the reactor would pass through 1: periods in ;ust 1 sec. The !ission ratehence the power would increase by a !actor o! e1:7 22,:::.

Imagine a reactor operating initially at 1 0 power, the system wouldreach a power o! 22,::: 0 in 1 sec in response to the change in ,i! it doesn


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ProblemProblem

A fast reactor assembly consisting of a homogeneous mixture of 239Pu and sodium

is to be made in the form of a bare sphere. The atom densities of these constituentsare N

F= 0.00395 x 1024for the 239Pu and N

s= 0.0234x1024for the sodium, while the

macroscopic absorption cross sections are 2.11 & 0.0008, respectively

Estimate the critical radius Rcof the assembly.

Answer but


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ProblemProblem

but

And from macroscopic transport cross sections


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Reactor KineticsReactor KineticsReactor with Delayed Neutrons

There are si3 groups o! delayed neutronssi3 groups o! delayed neutrons. To describe reactor /inetics

accurately one needs to consider all these groups. However, it will lead to acomplicated analysis. There!ore, in the !ollowing it is assumed, at themoment, that there is only one "rou%o! delayed that appear !rom a singlehypothetical precursor.

(onsider an infinite homogeneous thermal reactor

that may or may not be critical.Then in view o!,

Time-de%endent diffusion e&uationTime-de%endent diffusion e&uationcan be 'ritten ascan be 'ritten as

Note that the thermal flu( is inde%endent of %osition.In the above di!!usion e4uation=

n7 density o! thermal neutrons

ST7 density o! neutrons slowing down into thermal energy


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>ividing by

since

I! all the neutrons

were prompt, then?ut some !ission neutrons are delayed. This is 4uanti!ied by the !raction .

Then, only the 1 ) )!raction are prompt, and their contribution to STis,

(ontribution o! the delayed neutrons to the thermal source density is nothing

but the rate o! the decay o! the precursor C multiplied by the resonance

escape probabilityp &toms9

cm%


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Now the thermal source density can be written as

There!ore, the di!!usion e4uation becomes

Ne(t 'e need to determine the e&uation that 'ill determine the %recursorconcentration.This is achieved as !ollows= The rate at which !ission neutrons, o! boththe types, are given by

The rate at which delayed neutrons are produced is then

ince each delayed neutron results per precursor,the rate at which precursor !ormed is

The precursor also decays at usual rate


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There!ore, the e4uation which determines the %recursor concentrationis

These two coupled di!!erential e4uations

must be solved to determine C `

Now consider that a reactor is critical up to time t 7 :, and * 1.Then a

step change is made in so that the reactor is super critical or subcritical.It is re4uired to determine C as a !unction o! time a!ter t7 :.`

These e4uations can be solved by assuming solutions o! the !orm=

whereA C:are constants, and wis a

parameter to be determined.

ubstitution o! e3pression o! Cinto precursor e4uation gives

@ast three e3pressions when substitutedinto di!!usion e4uation yields


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Reactivity () of the infinite reactor

For a !inite reactor,

In terms o! reactivity,

Reactivity equation for one

group delayed neutrons.

+f reactor is su%ercritical, then reactivity is positive, and the reactor is saidto havepositive reactivity.

On the other hand, i! the reactor is subcritical, then reactivity is negative andthe reactor is said to have negative reactivity.

Aalues o! reactivity are restricted in the range


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This gives the relation between

reactivity and w, which in turn willde!ine solutions o! the >5.

The above e4uation must be solved to obtain the value o! w!or a given value

o! reactivity. This can be done by plotting #H o! the above e4uation as a

!unction o! w.

NOTE:NOTE: Three branches #oots are located by intersections!or a given value o! reactivity Two intersections !or a given value

o! reacivity. olution can then be written as

,ve reactivity,ve reactivity= w2 < w1, 2ndterm dies

with increasing time

-ve reactivity-ve reactivity= w2 < w1and again 2nd

term dies with increasing time.


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In case o! Bve reactivity, the !lu3 increasesas

hile in the case o! "ve reactivity, the !lu3 decreasesas

In both the cases,

#eciprocal o! w1is called reactor %eriodreactor %eriodor stable %eriodstable %eriod

Then the !lu3 behaviour can be written as

I! all the si3 delayed groups are considered, then the reactivity e4uation canbe written as

Now there will be seven roots !or

a given value o! reacivity.

However, as earlier, only the !irst root is more relevant, and the !lu3eventually approaches

#eactor period is again reciprocal o! w1


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The reactor period !or 2%*6 !ueled reactor

obtained using the si3 group reactivitye4uation !or negative and positive reactivitiesis shown in the !igure.

Now let us consider again that the value o!is varied !rom 1.::: to 1.::1. hat will bethe response o! the reactor to this changeC

The above change in the value o!

multiplication !actor implies lp7 1:"'

and reactivity 7 1:"%

.

The reactor period can then be directly read!rom the plot, which comes out to be

57 sec57 sec.


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&gain consider,

O# /T #TO#O# /T #TO#

The prompt neutrons li!etime, as noted earlier lp

is on the order o! 1:"Dsec.

Then the 4uantity wlp can be neglected. Then the reactivity e4uation !or a!ast reactor becomes,

For a !ast reactor !ueled with 2%*6, values o! period can be obtained !rom the

previous plot !or a curve lp7 :.


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Reactor KineticsReactor KineticsThe Prompt Critical StateThe Prompt Critical State

0ultiplication !actor as !ar as prompt neutrons are concerned is(1- )k.

There!ore, when (1- )k= 1,the reactor is said to beprompt critical.In this case the period is very short.

The reactivity !or the prompt critical state can be !ound by substituting

k7 19 (1- )into

?ut since 7 :.::E*, !or thermal neutron induced !ission o! 2%*6. The thermalreactor !ueled with 2%*6 becomes prompt critical by addition o! only about:.::E* or :.E*8 in reactivity.

The amount o! reactivity necessary to ma/e a reactor prompt critical is used to

de!ine unit o! reactivity called dollar. Note that the value o! varies !rom!uel to !uel, hence dollar is NOT an absolute unit.

e.g., a dollar is worth :.::E* value o! in reactivity !or 2%*6 !ueled reactorwhile a dollar is worth :.::2E in reactivity !or 2%%6 !ueled reactor.

The reactivity e4ual to One hundredth o! a dollar is called cent.


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Reactor KineticsReactor Kinetics Equal to ONEEqual to ONE

It is assumed that the precursor concentration is constant at its value in the

critical reactor. Then dC/dt7 :,there!oreince originally multiplication !actor 7 1

For


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Reactor KineticsReactor Kinetics Equal to ONEEqual to ONE

Then !lu3 4uic/ly assumes value o!

?ut,

Hence,

This e4uation used to see the e!!ect o! changing k, or in other wordschanging reactivity.

Two separate cases can be considered=one !or positive reactivity i.e. when increase in the power is re4uired,and the second !or negative reactivity.

One more case can also be considered 'hen

is NOT Equal to ONEis NOT Equal to ONE


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Temperature effects on ReactivityTemperature effects on Reactivity

The %arameters%arametersthat determine multi%lication factormulti%lication factordepend on temperature.depend on temperature.&nychange in the temperaturewill change k, and hence alters the reactivity o! the system.This in turn determines the reactor operation, and ultimately its sa!ety.

Temperature Coefficients

5!!ect o! temperature on the reactivity is 4uanti!ied in terms o!Temperature Coefficient of Reactivity, and is denoted as

?ut,

Tem%erature coefficient of reactivity is e&ual to the fractional chan"e ink%er unit chan"e in tem%erature.

3nit de"rees)3nit de"rees)-1-1

The algebraic sign o! the TCRdetermines response o! the reactor to the

change in temperature.


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ThreeThreedi!!erent cases can be discussed=

2 for %ositive TCRand 1 for ne"ative TCRCASE 1: +ve reactivityNote that kkis always positive. Then dk/dTis also positive. This implies thatthe increase in T'ill increase k.

Now, i! !or some reason the temperature o! the reactor increases. This

increases kk. The increase inkkwill increase the power,which increasesTT,which in turn will increase kkand so on.

There!ore, with positive T(#, rise in temperature will lead to ever increasingtemperature and power, until the reactor is eventually shut do'nby e3ternalintervention or it melts do'n.

CASE 2: +ve reactivityNow suppose the temperature o! the reactor decreases. This implies decrease in

kk, which will decrease the power. This, in turn, will decrease the temperature,!urther reducing kk, so on. Then reactor will eventually shut down.


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CASE 3: -ve reactivity This im%liesThis im%lies dk/dTdk/dTis alsois also ne"ativene"ative..

(onsider an increase in T, this will decrease kk which implies reduction inpower hence decrease in T. This returns the reactor to its original state.

On the other hand, i!Tdecreasesthen kincreases, which will increase power" increasing temperature. This returns the reactor to its original state.

rom all the three cases it is clear that a reactor 'ithrom all the three cases it is clear that a reactor 'ith $O/+T+5 T#$O/+T+5 T#isisinherentlyinherently unstableunstable 'hile a reactor havin" 'hile a reactor havin" N6T+5 T#N6T+5 T#isisinherentlyinherentlystablestable..

Increase in reactor power is re!lected firstin the rise in the !uel temp. In caseo! thermal reactor, temp. o! coolant and moderator rises when the heat is

trans!erred to these regions !rom !uel. One can then, de!ine Temp. (oe!!. !ordi!!erent. e.g.

The uel Tem%. oeff.is de!ined as the !ractional change in kper unitchange in !uel temp.

The 7oderator Tem%. oeff.is de!ined as the !ractional change in kper

unit rise in moderator temp.


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ince !uel temperature reacts immediately to the change in the reactor power,

the !uel temp. coe!!. is also /nown asprompt temperature coefficientand isdenoted as . Its value determines first res%onse of the reactor to anychange in either !uel temp. or reactor power.

For this reason, is the most important temperature coe!!icient as !aras reactor sa!ety is concerned.

$rom%t tem%. coeff. of most of the reactors is ne"ativedue to aphenomenon calledNuclear Doppler Effect.

&lthough the moderator tem%. coeff. is o! less important as compared toprompt. temp. coe!!., determines ultimate behaviour o! reactor inresponse to any change in !uel temperature.

It is desirable to have negative value o! to ensure stability duringnormal operation and during accident conditions.


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Fission Product PoisoningFission Product Poisoning

&ll fission %roduct absorbs neutronsfission %roduct absorbs neutronsto some e3tent, and their accumulation

tend to reduce value o! kk. ince absorption cross"section rapidly decreaseswith increasing energy, such fission product poisonsfission product poisons are o! greatest importancein thermal reactors.

Only e!!ect o! presence o! !ission product poisons on the value o! kis throughthe thermal utiliGation. Thus reactivity e4uivalent o! poisons in the reactor

which is previously critical is

For poisoned reactorFor poisoned reactor

In the absence of poisons, f0can be written as

With %oisonsWith %oisonsf becomes,


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Fission Product PoisoningFission Product Poisoning

The last e4uation can be written in a more convenient way by using

multiplication !actor !or unpoisoned reactor as !ollows=

Then, becomes

This e4uation will be used !or calculations o! !ission product poisoning.

M t i t t


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Xenon - 135Xenon - 135 Most importantfission product poison

Thermal absorption cross"section o! 1%*+e is 2.E* 3 1:Eb, which is 836

and is non"19!

ince 1%*Te decays rapidly, it can be assumed that 1%*I is !ormed ON@ in!ission. &ll these isotopes have di!!erent e!!ective yield which can be !ound!rom tabulated data.

For iodine:1%*I concentrationin atoms9cm%

5!!ective yieldatoms per !ission

For xenon:

M t i t t


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Then the e!!ective decay constant o! 1%*+e is

Equilibrium XenonEquilibrium Xenon

ince hal!"lives o! iodine and 3enon are so short that they 4uic/ly reach theire4uilibrium values. These values can be !ound by setting time derivative toGero.

M t i t t


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Temp. dependentparameter.

Aalue at 2: o(

Now reactivity e4uation becomes,

TWO Cases:TWO Cases:

(1)(1)

(2)(2)

M t i t t


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Maximum ValueMaximum Value

Linear increaseLinear increasewithwith

In the case o! a thermal reactor !ueled ON@ with 2%*6, and no resonanceabsorbers "

0a3imum value o! reactivity is=

'hich is about'hich is about 4 dollars4 dollars..

This is the ma(imum reactivity due to (enon e&uilibrium inThis is the ma(imum reactivity due to (enon e&uilibrium in 29:29:3 fueled3 fueledreactor.reactor.


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