John A. Bernard 5ubcritical Mulñplicaiion and Reactor 5ianup

Dr. John A. Bernard

MIT Nuclear Reactor Laboratory

John A. Bcmzrd Subcritical ivtuldplication znd Reactor Startup Page 2

A. Subcritica l Reacto r Behavior

— Neueon Sources

Out line

— Source - Detector Geometry

— NeuEon Life Cycle

— S ubcritical Muliiplicalion

— Prompi and Delayed Neulrons

— Reactivity

— Power-Period Relaiion

— Dynamic Period Equation

— Step and Ramp Reactivity Transients

— Coolant Temperature ( Moderator Coefficieni )

— Fuel Temperatwe ( Doppler Effect )

— Xenon

— Estimated Critical Posiiion

— Examples of Reac i o r Transients

— Laboratory Exercise

First Lecture

Second Lecture

Reacto r Physic s an d Operation

Nuclear fission reactors operate by maintaining a precise neunon balance. The reactor is in a critical condition if the number of neutrons created by the fission process equals lhe number that are either lost by leakage or captured within the reactor's structural materials. Neutrons produced from fission are called ”fast“ because they are traveling at v ery high speeds. Uranium-235, which is ie principal fuel in most reactors absorbs very few high- energy or fast neurons. In conaast, it will absorb, in large quantity, those neutrons that are moving slowly. Such neutrons are referred to as being "thermal.“ Hence, in order for the fission process to be maintained, it is necessary lhat the fast neutrons produced from fission be slowed down or thermalized. The need for the efficient thermalization of fast neutrons drives both the design and the operation of nuclear reactors.

Reacto r Operatin g Regimes

Reacior Operation is naditionally divided info ihree regimes. These are:

1 . Subcritical — Critical

a ) Subcriiical Muliiplication

2. Crilical — Poin t o f Addin g Heal

a ) Inhour @uation

b ) Dynamic Period Equaiion

3. Poin t o f Addi n Hear — Ho i Operating

a ) Temperature Feedback

b ) Doppler Effect

c ) Xenon Feedback

The same physical relations describe all free regimes. However, because different terms in those relations dominaie during each regime, the equations often look different. We will cover certain aspects of all three regimes with emphasis on subcritical multiplication, the dynamic period equation, and feedback mechanisms.

Subcritica l — Critical

— Crucial concepts are:

— Need for neulron sources

— Source-detector geometry

— Neuaon thermalization

— Subcritical multiplication

— '1/M' plots

Ne e d lo r Sourc e Neutron s

— It should be possible to monitor the neutronic condition of a reactor at all times, including when shutdown.

— Nuclear instnmenls ( power level and rate of change of power level ) musi be on scale prior to initiating a startup. Otherwise, the operator has no means of determining if his actions are having the intended effect.

— If a reactor core is brand new or if a reactor has been shutdown for many months, then the neutron population will be so low as to be undetectable. Small movements of the conrol devices can therefore resuli in large rates of changes — rates that are so rapid that power can rise to the level where fuel damage occurs before the nuclear safety system is capable of responding.

— The installation of a neuFon source ensures safety by keeping all instnments on scale.

Neutro n Sources

l . Photo- Ne uiro n

0 2 D o n + i H

Note: Fission products provide lhe gamma rays. So, the reactor musl have a power history for ihis source to be effective. The needed fission products decay wilhin 2-3 months.

2. Plutonium- Bervllium

g 9 P J —i 2 He + 9 U

2 He + Be —› ' Q C + l

CAUTION: PuBe sources are doubly encapsulated in steel. Hence, heat iransfer is poor and PuBe source must not be left in a feactor if power exceeds a few hundred Watts.

N eutro n Source s ( cont. )

3. Antimonv-Bervllium

n + ’ 2 3Sb —i ' 2 4 Sb

' 2 4 Sb —i 0

O 7 + JBe —i l + 2 ( He )

Note : Musl have radioactive antimony as initial condition.

Source-Detecto r Geometry

— Photoneutrons are homogeneous because neutrons are produced throughout lhe entire volume of the core. The other types of sources are discrete entities and care must be taken to ensure proper source-detector geometry.

— A discrete source should be placed in the center of the reactor core with fuel surrounding it. The detectors should be located beyond the fuel. Thus:

D

D D

D

— Power reactors are required to have four detectors, one in each quadrant.

Sxa les of Incorrect Source-Detector GeometrY

Configuratio n

Pr‹iblem

Both source and detector are al core cenler. Hence, deleclor regislers only source neutrons.

Both source and detector are external to core. Again, the deteclor only registers source neutrons.

Source is exierior lo core. Hence, only a small fraction of lhe fuel is exposed io the source neulrons.

Courtesy of Brookhaven National Laboratory.

Critical k ,ff

n $ ' e p fL$ L

‘f

Neutro n Lif e Cycle

Definition s o f Neutro n Lif e Cycl e Factors

Total Number of Fast Neurons Produced from Fast and Thermal Fission Number of Fast Neulrons Produced from Thermal Fission

L Tota l Numbe r o f Fas t Neuron s Escapin g Leakag e

f * Total Number of Fast Neurons Produced from Fast and Thermal Fission

_ Tota l Numbe r o f Thermalize d Neutron s

* Total Number of Fast Neurons Escaping Leakage

L = Tota l Numbe r o f Therma l Neutron s Escapin g Leakage

Toial Number of Thermalized Neurons

Definition s o f Neutro n Llf e Cycl e Factors ( cont. )

Thermal Neurons Absorbed in Fuel

Total Number of Thermal Neutrons Escaping Leakage

Therma l Neuron s Caplure d i n Fue l Whic h Caus e Fission

Of

Thermal Neutrons Absorbed in Fuel

Numbe r o f Fas t Neuron s P ro duce d fro m Therma l Fission Thermd Neurons Absorbed in the Fuel

Of the above factors, the reactor operalor can aller 'F by changing lhe control rod position or by adjusting the soluble poison content. The leakage terms also vary during routine operation whenever coolant temperature changes. The other terms are fixed by the fuel type.

Cor e Multiplicatio n Factor

l . li is useful to define a 'core mulliplicaiion factor' which is denoted by the symbol 'K' and which is the product of ihe six factors that define the neutron life cycle. Thus,

K = tL f pL fr}

2. The above expression, which is called ihe 'six-facior formula, has physical meaning:

Neuiron s Produce d fro m Fissio n

K= Neurons Absorbed + Neutron Leakage or

K = Numbe r Neutron s i n Presen t Generatio n Number Neutrons in Preceding Generaiion

K= I \ k p 1 1 2 » f l 3 when n is ihe number of

n o n n 2 neutrons in each generation.

3. If K is unity, the reactor is critical.

4. If we know th e K-value for a reactor core, we can determine the rate of change of its

neutron population. This is most useful in reactor starNps.

		Neutrons From
Generation	Source	Multiolication	Toial
0	100	0	100
l	100	60	160
2	100	b0 + 36	196
3	100	60 + 36 + 22	218
4	100	60 + 36 + 22 + 13	231
5	100	60 + 3b + 22 + 13 + 8	239
	100	60 + 3b + 22 + 13 + 8 + 5	244
7	100	b0 + 36 + 22 + 13 + 8 + 5 + 3	247
8	100	60 + 36 + 22 + 13 + 8 + 5 + 3 + 2	24 9
9	1 00	60 + 36 + 22 + 13 + 8 + 5 + 3 + 2 + 1	250
10	100	b0 + 36 + 22 + 13 + 8 + 5 + 3 + 2 + l + 0	250

All neucon quantities have been rounded to the nearest unit. Note that the incremental increase to the total population from the first generation is zero in the tenth generation. Hence, the system has reached equilibrium.

Generaiio n # Sourc e Neutrons Neutron s fro m Multiolication

0 0 0

l 2	***0 S O	KS 0 K S 0 + K 2 S 0
3 4	0 S O	K 5 + K 2 S 0 + K 3 S K S 0 + K 2 S 0 + K 3 S + K 4 S 0
	0	K 5 0 + K 2 S 0 + K3s 0 + K 4 S 0 +	+ K n S 0

Mathematic s o f Subcritica l Multiplication ( cont. )

So, afier n generations, the neutfon populalion would be:

Tolal Neutrons = S 0 + KS 0 + K 2 S 0 + K 3 S 0 + K 4 S 0 + + K n S 0

S 0 ( 1 + K + K 2 + K 3 + K 4 + + K R )

I - K

Note : The fact that the series l + K + K 2 + K 3 + K 4 + + K R does equal l/ ( l-K ) can be proved mathematically for values of K ihal are less lhan l.

— We now have an expression that may be used to calculate the equilibrium neutron level in a subcritical reactor.

— Of extreme importance to reactor startups is that lhe total neutron population in a subcritical fissile medium exceeds the source level by a factor of 1/ ( l-K ) or UM. This process is called subcritical multiplication.

— The following should be noted:

a ) The subcritical multiplication formula does NOT allow calculation of the time required for criticality.

b ) As lhe multiplication factor, K, approaches 1.0, the number of generations and hence time required for the neutron level to stabili2e gets longer and longer. This is one of the reasons why it is important to conduct a reactor startup slowly. If it isn't done slowly, the subcritical multiplication level won't have time to attain equilibrium.

c ) The equilibrium neutron level in a subcr i tical reactor is proportional lo the initial neutron source strength. This is why it is important to have neutron count rales above a certain minimum before conducting a stuNp.

d ) The formula is only valid while subcritical.

— How can we apply the subcritical multiplication formula? We can measure neutron counts and source strength. The latter is merely the neutron counts with the reactor shutdown. ¥Ience, we can calculate the multiplication factor, K, and thereby estimate how close a reactor is to criticality.

— We derived the relation:

Cou n ts = S@ ( I-K )

— Rearrange the above to o btain:

( l-K ) Sp/Counts

— To make use of lhis relation, plot inverse counts ( Sp/Counts ) on the veriiñal axis and control rod position or poison concentration or fuel loading on the horiz o ntal axis. The result is called a ’1/M' plot where Nt siands for multiplication. The point where ihe plot is exirapolated to cross the horizontal axis is where K equals l and the reactor is critical.

' l/M ' Plot

Source/Coun t Rate

1.o

0.77

0.51

0.24

0.125

'1/M ' Plot ( cont. )

If the data from this example is plotted such that the vertical axis is ( Source/Count Rate ) or Sp/CR, we get:

CR

0 1 2 3 4 5 6 7

Blade Height

8 9 10

Criticality is expected at 8.0”. Actual plots are usually not linear. Instrument noise and

improper source-detector geometry cause a non-linear response.

Critica l

— Crucial concepts are:

— Prompl and delayed neutrons

— Importance of delayed neutrons

— Power-period relation

Reactivity

— Dynamic period equation

— Step and ramp reactivity transients.

— The 'point of a d ding heat’ is the power level above which a change temperature is observed following a change of power. Reactors may be critical at a few Watts or even less. At such low powers, no c h ange in temperature results because of the core's heat capacity. In general, there are no reactivity feedback effects below the point-o f- ad d ing heat.

Pro m t an; l DelaYe d Neutrons

The reactor multiplication factor, K, has been defined as the ralio of neulrons in one generation to Ihose in the immediately preceding generalion. Combining this definition with a physical understanding of Ihe neutron life cycle allowed us Io write an equation that predicted the equilibrium neutron count rate in a subcritical reactor. Unfortunately, lhat relationship is not valid for a critical or supercritical reaclor. We necd to develop a means of describing both the neutron level and its rate of change in a supercritical reactor. The first thing that we should do is Io examine the fission process and determine what lypes of neutrons are produced. The fission of a U-235 nucleus normally yields two fission fragments, an average of 2.5 neulrons, and an assortment of beta particles, gamma rays and neutrinos. The neutrons that are produced directly from the fission event are referred to as rop ;npI because they appear almost instantly. Most of the neulrons produced in a reactor are prompt. However, certain fission fragments, which are called precursors, undergo a beta decay to a daughter nuclide that -then emits a neulron. Neutrons produced in this manner are referred to as delayed. The delay is the time that must elapse for the precursor to underg o its beta decay. Delayed neutrons constitute an extremely small fraction of a reactor's lotal 'neutron population. Nevertheless, they are crucial Io the safe operation of a reactor during power tra n sients.

Thermal Neutron		Uranium

Delayed Neutrons		Daughter Nuclides

Im D ortanc e o f Delaye d Neutrons

— The time required for a prompi neutron to be born, thermalize, and cause a fission is on the order of l • l0 ‘ 4 s. This is too rapid for human or machine conlTol.

— Delayed neutrons have an average lifelime of 12.2 s.

The fraction of neutrons lhat are delayed in a typical light-water reaclor is 0.0065. This quantity is denoled by the Greek leiter § and is called Beia.

— Assume lhat a reactor has 100,000 neuirons present. The lifetime of an 'average' neutron is iherefore:

( # prompt n' s ) ( prompi lifetime ) + ( # delayed n's ) ( delayed lifetime )

tolal # neutrons

07 ( 99350 ) ( 0.0001 ) + ( 650 ) ( 12.2 )

ioo,ooo

or 0.079 s

Thus delayed neutrons lengthen the average neuson lifeiime and result in a controllable reaclor.

Reactivit y

— Reactivity is a measure of the departure of a reactor from criiicalily. It's malhemalical definition is:

K — l K

where K is ihe reactor's multiplication faclor.

— If the reactivity is negative, the reactor is subcritical. Conversely, if it is positive, the

reactor is supercritical. If the reactivity is zero, the reactor is exacily critical.

— Reactivity may be ihought of a the 'fractional change in the neutron population per

neutron generation.'

— Reaciivity is a global property of a reactor. Nevertheless, it is common practice to speak of the reaciivity worth of a control rod or of ltte soluble poison. Wilhdrawing a rod or dJuting the poison is said to 'add reactivity.'

— We noled earlier that there are lwo kinds of neutrons: prompt and delayed. The latter are produced on a time scale that is controllable by humans and instruments. Thus, it is esseniial that reacior kansients always be conducted in a manner such that the delayed neurons are the rate determining factor.

— The fraction of delayed neutrons is 0.0065. Therefore, the amount of posilive reactivity present in a reactor should never be all o wed to exceed some small percent of the delayed neutron fraction.

— Reactivity, being dimensionless, has no units. Bui, it is common lo measure it relative

to ihe delayed neuoon fraction. We say that:

1 Beta = 0.0065 AK/K

Reactivity is sometime also measured in dollars and cents with 1 Beta = $1 = 100 cenis.

— The following table illustrales lhe importance of limiting reactivlty additions. Shown are Ihree cases, all with the same initial condition: reacior critical with a population of 10,000 neurons.

— For the first case, no change is made. One generalion later there are 9935 prompt neutrons and 65 delayed ones. Criticaliiy can NOT be maintained wilhou t the delayed neuirons. Hence. they are the rare-determining step.

— For the second case, we add 0.500 Beta of rea ct iv i ly. This corresponds to ( 0.0065 AK/K ) ( 0.50 ) or 0.00325 AKJK or 33 neutrons in the first generation. Thus, afler one generation there are 9968 prom pt neutrons and 65 delayed ones for a total of 10,033. The delayed’ neurons are still controlling because it takes 100,000 neulrons to stay critical and there are only 5968 prompt ones.

— For the third case, we add 1.5 Beta of reactivity. This corresponds to ( 0.0065 ) AKJK ( 1.5 ) or 0.009 AK/K or 98 neurons in the first generation. Thus, after one generation there are 10,032 prompt neutrons and 66 delayed ones. There are more than enough

prompt neutrons to maintaln criticality. The prompt neutrons ue consolling with their

10- 4 s life cycle.

Initia l Condition InitiaLPonalaiion Critical 10.0tD neutrons

9,935 prompl

65 delayed

o¥a Ax1

or 0.0 Ben

or 0 neutrons

On e Geaeratlo H tRf

10.0 tXl

9,935

65

Condition

Steady-State; Not critical on prompt neutrons alone

10.0 tD neutrons 9,935 prompt

65 delayed

0.325% DK/K 10,033

or 0.5 Ben 9,968

or 33 neutrons 65

Supercritical; Not critical on prompt neutrons alone

l0,0tD neutrons

0.98 &» DK/K

10.098 Power Runaway;

9,935 prompl

or 1.5 Ben

or 98 neutrons

10,032

a

Critical on prompt neutrons alone

Power-Perio d Relation

— Reactivity is not direcXy measwable and hence most reactor operating procedwes do not refer to it. Instead, they specify a limiting rate of power rise, commonly called a 'reactor period.'

— Reactor period is denoted by the Greek lener, i, and is defined as:

v z n ( t ) / ( dn ( t ) / dt )

where n ( t ) is the reactor power. Thus, a period of infinity coaesponds to the critical condition.

— The relation between power and period is:

P ( t ) = P e"’

where P ( t ) is the power level, Pg is the initid power, e is ie exponentid, and t is

Suixnitical Multiplication and Reactor Startup

Examnle s o f Power-Perio d Relation

1. th

d 100

s and the initial power is 10% of rated. How long before

P ( t ) = P O e *

l009o=l0% e'* °°

ln ( l&/10 ) = I J 100 230 s = t

2. Suppose the reaclor period is equal

faclor would power rise in 1.0 ms?

to lhe prompl neuron lifetime of 1 • l 0 - 4 s. By what

P ( t ) = P e t

P / ( t ) J Pp = e 0 3 “

= 22,026

The reactor is uncon o o ll able.

3. Repeai problem fl2 with a period of 0.079 s. The answer is 1.013. Hence the value or

delayed neurons.