design requirements – reactor...

Design Requirements – Reactor Physics

22.39 Elements of Reactor Design, Operations, and Safety

Fall 2005

George E. Apostolakis Massachusetts Institute of Technology

Department of Nuclear Science and Engineering 1


Principal Design Functions Involving Reactor Physics

• ¾

production over the core life ¾ Depend on moderator-to-fuel ratio, core geometry, location and type of

reactivity control, fuel element design • Reactivity and control analysis (safety)

¾ Must control excess reactivity in initial fuel loading ¾ Describe short-term reactivity changes (and reactor kinetic behavior);

reactivity coefficients. • Depletion analysis (economic performance)

¾

Core criticality and power distribution Are space and time dependent because of fuel burnup and isotope

Monitor fuel composition and reactivity as a function of energy removal


The Big Picture

The Multigroup Diffusion Equations:

g

S

)(D tv

1

ext g

G

1g gfggg

G

1g g

gtggg g

g

' '''

' ''

=

+ϕΣνϕΣ=

=ϕ ∂

∑∑ ==

Group constants:

∫

∫

−

−

ϕ

ϕΣ

≡Σ

g

g E

E

E

E t

tg

)(dE

)()E(dE

EgEg-1Eg-2 Eg+1

Group g

2 MeV

Group g’

G ,...,2 ,1

t , r

g sg χ +

Σ + ϕ ∇ • ∇ − ϕ ∂

1 g

1 g

t ,E , r

t ,E , r

1 eV


Power Distribution

• Problem: The group constants depend on the flux itself. • Criticality: Set and equal to zero. •

¾

5 eV,

¾ This fine spectrum calculation may involve as many as 1000 groups. ¾

coarse group calculation with spatial dependence. • LWRs: Usually three fast groups and one thermal. • Fast Reactors: As many as 20 groups.

tv 1 g

g ∂

ϕ ∂ ext gS

E 1)E( ∝ϕ

)E()E( χ∝ϕ

Perform two multigroup calculations: Treat spatial and time dependence very crudely and calculate the intragroup fluxes relying on models of neutron slowing down and

thermalization, e.g., assuming that for energies between 1

eV and 10 in the high-energy range, and proportional to the Maxwellian distribution for thermal energies.

These intragroup fluxes are, then, used to calculate the group constants for a


Two-Group Criticality Calculation

1222

2211111

D

][k 1D

ϕϕ

ϕϕ=ϕ

)r()r()r()r( 2211 ψϕ=ϕψϕ=ϕ

)()( k 2

2

2 2

1 2

1

1 +Σ

Σν

+Σ

Σ+

+Σ

Σν =

Assuming that

We get the (effective) multiplication factor

For criticality: k = 1

0)r()r(B)r( s 22 =ψ=ψ+ψ∇

12s 2 a

2 f 1 f 1 R

Σ = Σ + ϕ ∇ • ∇ −

Σ ν + Σ ν Σ + ϕ ∇ • ∇ −

B D B D B D 2 a

2 f

1 R

12s

1 R

1 f

, 0

More on the Multiplication Factor

⎡ Σν 1f ⎤ ⎡ Σ 12s Σν 2f ⎤ k = ⎢ 2 ⎥

⎢Σ 1R

1 + BD 2 ⎥⎥⎦

+ ⎢⎢⎣ (Σ 1R + BD 2) (Σ 2a + BD 2) ⎥⎦⎣ 1 1 2

k1: due to fissions in the fast group

k2: due to fissions in the thermal group

⎛ Σ 12s ⎞ν 2 ⎜⎜⎛ Σ 2f ⎟

⎞ ⎜ ⎜⎝ Σ 1R ⎠

⎟ ⎟⎝ Σ 2a ⎠

⎟ ==k2 ⎛ 2 ⎞ ⎛

⎜ ⎜1 + D1 B ⎟ ⎟ ⎜⎜ 1 +

D2 B2 ⎞⎟⎟⎝ Σ 1R ⎠ ⎝ Σ 2a ⎠

f = 2 2 = η 22 Pp η 22 pPf 2NL1NL( 1 + BL 2 ) ( 1 + BL 2 )1 2



The Six-Factor Formula

⎟ ⎟ ⎠

⎞ ⎜ ⎜ ⎝

⎛

Σ

Ση F22

12 1

DL Σ

≡

( ) 22 11

1P +

≡

absorbed in the fuel

Diffusion area for fast neutrons

Nonleakage probability for fast neutrons

⎟ ⎟ ⎠

⎞ ⎜ ⎜ ⎝

⎛ Σ

Σ≡p

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛

Σ

Σ≡

F f

thermal neutron in the fuel

( ) PPkPPpfPk ∞=εη=εη=

⎟ ⎟ ⎠

⎞ ⎜ ⎜ ⎝

⎛ +

2

1 k k1 Fast fission factor

ν ≡ 2 a

2 f

1R

1NL B L

Average number of neutrons produced per thermal neutron

1 R

12 s Resonance escape probability

2 a

2 a Thermal utilization: conditional probability of absorption of a

2 NL 1NL 2 NL 1NL th th 2 NL 1NL 2 2 P p f

≡ ε


Comments

thη

thf

ε

233U: 2.29, 235U: 2.07, 239Pu: 2.15, natural uranium: 1.34, enriched uranium: 1.79. 238U,

About 0.9 for natural uranium.

p

About 1.05 for natural uranium.

1L Water: 0.052 m, heavy water: 0.114 m, graphite: 0.192 m

2L graphite: 0.54 m

B2 Typically less than 10 m-2, therefore PNL1 > 0.97 and P > 0.99 for H2O

Increases initially as Pu is produced from decreases later as fission products are produced

Larger as absorptions in nonfuel material decrease.

About 0.70 for homogeneous mixtures, 0.9 for heterogeneous mixtures, increases as the ratio of moderating atoms to fuel atoms becomes large.

Water: 0.027 m, heavy water: 1.0 m,

NL2


Time Dependence

• To study the time-dependence of the flux we have to solve the multigroup

• There are two time scales: ¾

changes ¾

buildup.

equations in slide 3 augmented to include the equations for delayed neutrons.

Short-term changes (seconds) due to temperature effects and external deliberate

Long-term changes (hours or more) due to fuel depletion and fission-product


Delayed Neutrons

Figure by MIT OCW.

87Br (55 sec)

87Kr

En

86Kr (stable)Neutron

B.E.

β- (~70%)

β- (~30%)


Delayed Neutron Precursors

t1/2(s)

λi (s-1) βi

βi/λi(s)

22.7

6.22

2.30

0.61

0.23

Total

Decay Constants and Yields of Delayed-Neutron Precursors in Thermal Fission of Uranium-235

0.0065 0.084

55.7

0.0305

0.111

0.301

1.1

3.0

0.0124

0.00142

0.00127

0.00257

0.00075

0.00027

0.000215

0.0466

0.0114

0.0085

0.0007

0.0001

0.0173

Figure by MIT OCW.


Precursor Data

Figure by MIT OCW.

Approximate Half-Life(seconds)

55

23

6.2

2.3

0.61

0.23

5.7 x 10-4

19.7

16.6

18.4

3.4

2.2

34.6

31.0

62.4

18.2

6.6

18.2

12.9

19.9

5.2

2.7

5.2 x 10-4 2.1 x 10-4 0.25

0.46

0.41

0.45

0.41

-

0.0061

2.93

0.0020

0.0158

2.42

0.0065

0.0066

2.49

0.0026

Total Delayed

Total Fission Neutrons

Fraction Delayed

Number of Fission Neutrons Delayed / Fission

U-233 U-235 Pu-239

Energy (MeV)


Point Kinetics

• Recall (slide 5):

• Local perturbations, e.g., by moving the control rods, leads to a readjustment of that is usually slight and happens in a few milliseconds. Then, the readjusted shape rises or falls “as a whole” (harmonics) depending on whether the perturbation increased or decreased k.

• Point kinetics allows us to investigate the level (or average) flux assuming that the shape does not change appreciably.

• We average over all energy groups and write the neutron density as

• n(t) is the total neutron density or the total power

• Using this equation in the space- and time-dependent equations and including delayed neutrons leads to

)r()r( )r(

2

1

2

1 ψ⎟⎟ ⎠

⎞ ⎜⎜⎝

⎛ ϕ

ϕ =⎟⎟

⎠

⎞ ⎜⎜⎝

⎛ ϕ

ϕ 0)r(,0)r(B)r( s 22 =ψ=ψ+ψ∇

)r(ψ

)r()t(n)t,r(n ψ=

)t(n)t,r(dVw)t(Power G

1'g V 'g'fg'fg ∝ϕΣ= ∑ ∫

=


)1(v 1

)t(k 22 a +Σ

≡≅ AA Mean generation time between birth of a neutron and absorption inducing fission

A 10-3 to 10-4 for thermal reactors; 10-7 for fast reactors

B L ≡ Λ

Prompt neutron lifetime between birth of a neutron and absorption;

Point Kinetics Equations

6dn ( t ) =ρ ( t ) β − n( t ) + ∑ λ iC i ( t )

dt Λ 1 dC i ( t )

=β i n ( t ) λ − iC i ( t ) i = 6 ,..., 2 , 1

dt Λ

ρ ( t ) ≡ k ( t ) − 1 Reactivity

k ( t )


Importance of Delayed Neutrons

⎟ ⎠ ⎞

⎜ ⎝ ⎛≡⎟

⎟ ⎠

⎞ ⎜ ⎜ ⎝

⎛ −∝

T t expt1k exp)t(n

A 1kT

− ≡ A

∑ ⎥ ⎦

⎤ ⎢ ⎣

⎡ +

λ β+=

6

1 i i

1)1( AAA

The reactor period is less than 10-3 s for thermal reactors, if delayed neutrons are neglected. control possible.

Reactor period without delayed neutrons1kT

−≡ A

Reactor Period

β −

It is about 0.1 s with delayed neutrons, thus making reactor

Prompt neutron lifetime with delayed neutrons


Reactivity Feedback

• •

themselves involve the atomic number densities of the materials:

• The atomic density depends on the power level because: ¾

distribution ¾

• We write

0 for which is zero. Change in reactivity due to inherent feedback mechanisms.

)()(N)( σ•=Σ

]P[)t()t( fext δρ+δρ=ρ

)t(extδρ

]P[fδρ

ρ

The reactivity depends on the neutron density (or power level) itself. This is due to the fact that k depends on macroscopic cross sections, which

Material densities depend on temperature, which, in turn, depends on the power

The buildup of poisons and burnup of fuel (long-term effect).

External reactivity from some reference power level P

t , r t , r t , r


Power Coefficient of Reactivity

⎟ ⎟ ⎠

⎞ ⎜ ⎜ ⎝

⎛ ∂

∂ ⎟ ⎟ ⎠

⎞ ⎜ ⎜ ⎝

⎛

∂ =

ρ≡α ∑ P

T TdP

d j

j j P

∫ α≡ FP

0 PPD dP Power Defect:

soluble boron. It’s less than about 0.05.

To be determined by T-H analysis

Safety requirement: 0P <α

ρ ∂

ρ ∆

Temperature coefficient of reactivity for region j

Total reactivity change from hot zero-power state to hot full-power state; compensated by control-rod insertion and

Average temperature of region j


Isothermal Temperature Coefficients of Reactivity

• Assume a uniform reactor with the temperature uniform throughout the reactor.

( ) PPkPPpfk ∞=εη=

dt dP

P 1

dt dP

P 1

dt dk

k 1

dt dk

k 1

dt dk

k 1

dt d

2 ++=≈= ρ ∞

∞

( )22NL 1

1P +

≡

thη

ε 235U, larger (but still small) for 239Pu.

Negligible effect

p Main

.

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛

Σ

Σ≡

F f

Main . Could be positive.

2 NL 1 NL 2 NL 1 NL th th

2 NL

2 NL

1NL

1NL

B L Decreases with temperature because of thermal expansion; for large thermal reactors, a negligible effect.

Usually negative effect; small for

Decreases due to Doppler broadening of the absorption resonances; not too important for homogeneous reactors, very important for heterogeneous reactors. contributor to the prompt (fuel) temperature coefficient

2 a

2 a Cross sections depend on energy spectrum, hence on moderator temperature. contributor to the delayed (moderator) temperature coefficient


Temperature Coefficients

Figure by MIT OCW.

BWR PWR HTGR LMFBR

Fuel-Temperature CoefficientDoppler (pcm/oK)

Isothermal Temperature CoefficientCoolant Void (pcm / %void)Moderator (pcm/oK)Expansion (pcm/oK)

Temperature Defect (%∆k/k)Power Defect (%∆k/k)Xe Worth (%∆k/k)Sm Worth (%∆k/k)

2.0-3.01.5-2.5

2.60.7

-4 to -1

-200 to -100-50 to -8

~0

2.0-3.01.5-2.5

2.60.7

-4 to -1

0-50 to -8

~0

0.50.80.00.0

-0.6 to -2.5

-12 to +20

- .92

0.74.03.30.5

-7

0+1.0~0


General Design Criteria (10 CFR 50 Appendix A)

•

structures, systems, and components important to safety; that is,

assurance that the facility can be operated without undue risk to the health and safety of the public.

•

other units.

The principal design criteria establish the necessary design, fabrication, construction, testing, and performance requirements for

structures, systems, and components that provide reasonable

The General Design Criteria are also considered to be generally applicable to other types of nuclear power units and are intended to provide guidance in establishing the principal design criteria for such


General Design Criteria 27 and 28

• Criterion 27--Combined reactivity control systems capability. The reactivity

the core is maintained.

• Criterion 28--Reactivity limits. The reactivity control systems shall be designed

to assure that the effects of postulated reactivity accidents can neither (1)

addition.

control systems shall be designed to have a combined capability, in conjunction with poison addition by the emergency core cooling system, of reliably controlling reactivity changes to assure that under postulated accident conditions and with appropriate margin for stuck rods the capability to cool

with appropriate limits on the potential amount and rate of reactivity increase

result in damage to the reactor coolant pressure boundary greater than limited local yielding nor (2) sufficiently disturb the core, its support structures or other reactor pressure vessel internals to impair significantly the capability to cool the core. These postulated reactivity accidents shall include consideration of rod ejection (unless prevented by positive means), rod dropout, steam line rupture, changes in reactor coolant temperature and pressure, and cold water


Standard Review Plan: 4.3 Nuclear Design 2

include:The areas concerning reactivity coefficients

The applicant's presentation of calculated nominal values for the reactivity coefficients such as the moderator coefficient, which involves primarily effects from density changes and takes the form of temperature, void, or density coefficients; the Doppler coefficient; and power coefficients.


• The poisoning effect is insignificant for fast reactors.

• Most important products: with and with

• causes.

• The thermal utilization affected by the poison.

Xe135 54

Sm149 62

2226X a m−−=≅σ

2244S a m−−=≅σ

⎟ ⎠ ⎞

⎜ ⎝ ⎛ −=

−−

− k1k 1

k 1k1'

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛

Σ

Σ≡

F f

a

aP

a

aPa1f1k1k 1

Σ

Σ

Σ

Σ−= ⎟

⎠ ⎞

⎜ ⎝ ⎛ −≅ ⎟

⎠ ⎞

⎜ ⎝ ⎛ −

a

aP Σ

Σ

Core Composition Changes – Fission Product Poisoning

Some fission products have large thermal absorption cross section.

We measure the impact of a poison by calculating the reactivity decrease it

is the only factor that is appreciably

10x3 b 10x3

10 x 3 b 10 x 5

= ρ − ρ = ρ ∆ 'k 'k

'k

2 a

2 a

' f ' k − =

Σ + = ρ ∆ − = ρ ∆

Xenon Poisoning

222X m−−=σ

Fission

135I 135Xe

γI = 0.061 γX

λI = 2.9x10-5 s-1

λX = 2.1x10-5 s-1

136Xe

135Cs

10 x 0 . 3

= 0.003

a

∂ I( t , r ) Σ γ = f ϕ ( t , r ) λ − I I ( t , r )

∂ t I

∂ X ( t , r ) γ = X Σ f ϕ ( t , r ) λ + I I ( t , r ) λ − X X ( t , r ) σ − X ϕ ( t , r )X ( t , r )

∂ t a

I ∞ =γ I Σ f ϕ 0 X ∞ =

( γ I + γ X )Σ f ϕ 0λ I λ X σ + a

X ϕ 0

I f 2− = ρ ∆ σ a

X ( γ + γ X ) ϕ Σ 0 − = ρ ∆ 023.0 for ϕ 0 = 1017 n / s m (λ X σ + X ϕ 0 )a



Xenon and Reactor Shutdown

)(X)(I t

)(X

)(I t

)(I

XI

I

λ= ∂

∂

= ∂

∂

A reactor operating at a flux of 2x1018

n/m2s will have a negative insertion of reactivity of about -0.33, a sizable amount.

Reactor Engineering, 1994

t , r t , r t , r

t , r t , r

λ −

λ −

Graph removed for copyright reasons.

Source: Glasstone & Sesonske, Nuclear Chapman & Hall,

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