Reactor Physics – D esign Parameters for PWRs

22.39 Elements of Reactor Design, Operations, and Safety Lecture 5

Fall 2006

George E. Apostolakis Massachusetts Institute of Technology

General Aspects

• Design Objective

 Safe, reliable, economical operation at the rated power level over the core lifetime.

• Interaction with other disciplines

 Thermal considerations determine core size and geometry.

 The length of time a fuel element can utilized is determined by its ability to withstand radiation damage and thermal/mechanical stresses.

• The design process is iterative drawing heavily on past experience.

Principal Design Functions Involving Reactor Physics

• Core criticality and power distribution

 Are space and time dependent because of fuel burnup and isotope production over the core life

 Depend on core enrichment, moderator-to-fuel ratio, core geometry, location and type of reactivity control, fuel element design

• Reactivity and control analysis (safety)

 Must control excess reactivity in initial fuel loading

 Allocate this reactivity to movable control rods, soluble neutron poisons in the coolant (“chemical shim”; boron), and “burnable poisons” or “mechanical shim” (gadolinium, borosilicate glass).

 Describe short-term reactivity changes (and reactor kinetic behavior); reactivity coefficients.

• Depletion analysis (economic performance)

 Monitor fuel composition and reactivity as a function of energy removal

The Big Picture

The Multigroup Diffusion Equations:

Group g’

Group g

2 MeV 1 eV

1   g

   D      ( r , t ) 

E g- 2

E g- 1 E g

E g+1

v g  t

g g tg g

G

 

g '  1

 sg ' g  g '

  g

G



g '  1

 g '

 fg '

 g '

 S ext

g  1 , 2 ,..., G

Group constants:

 tg

E g  1

 dE  t ( E )  ( r , E , t )

 E g E g  1

 dE  ( r , E , t )

E g

Power Distribution

• Problem: The group constants depend on the flux itself.

• Criticality: Set

1   g

v g  t

and

ext g

equal to zero.

• Perform two multigroup calculations:

 Use a library code to obtain cross sections and average them by treating spatial and time dependence very cr udely. Calculate the intragroup fluxes relying on models of neutron slowing down and

thermalization, e.g., assu ming that

 ( E )  1

E

for energies between 1

eV and 10 5 eV,

 ( E )   ( E )

in the high-en ergy range, and prop ort i onal to the

Maxwellian distribution for thermal energies.

 This fine spectrum calculation may involve as man y as 1000 groups.

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

coarse gr oup calculation with spat ial dependence. Iteration is possible.

• LWRs: Usually three fast groups and one thermal.

• Fast Reactors: As many as 20 groups.

Two-Group Criticality Calculation (Bare Homogeneous Reactor)

  

D       1 [  

     ]

1 1 R 1 1

k 1 f 1 1 2 f 2 2

 R 1

  

  a 1   s 12

D 2   2

  a 2  2

  s 1 2  1

Assuming that

 1 ( r )

  1  ( r )

 2 ( r )

  2  ( r )

We get the

 2  ( r )  B 2  ( r )

 0 ,

 ( r s )  0

(effecti ve)

k   1  f 1 

 s 12

 2  f 2

multiplication

factor

 R 1

 D 1 B 2

(  R 1

 D 1 B 2 )

(  a 2

 D 2 B 2 )

More on the Multiplication Factor

k    1  f 1   

 s 12

 2  f 2 

  

  D B 2 ( 

 D B 2 ) ( 

 D B 2 ) 

  R 1

1     R 1 1

a 2 2  

  s 12    f 2 

    2   

k 2   R 1   a 2  

 D 1 2   D 2 2 

 1 



B 

R 1 

 1 



B 

a 2 

  p

   2 f 2

   2 f 2 p P NL 1 P NL 2

The Six-Factor Formula

k   2 f 2 p  P NL 1 P NL 2

   th f th p   P NL 1 P NL 2

 k  P NL 1 P NL 2

  f 2 

 2   2  F 

Average number of neutrons produced per thermal neutron

  a 2 

 

  F 

absorbed in the fuel

Thermal utilization: conditional probability of absorption of a

f 2 

 a 2 

  a 2 

thermal neutron in the fuel

L 2 

P NL 1 

D 1

 R 1

1

2 2

1

Diffusion area for fast neutrons

Nonleakage probability for fast neutrons

  s 12 

p  





 R 1 

Resonance escape probability

 k  

  1  f 1   

 D B 2  

   1  1    1  

  a 2 2  

Fast fission factor

 k 2     2  f 2    s 12  

   

Comments

 th

f th

233 U: 2.29, 235 U: 2.07, 239 Pu: 2.15, natural uranium: 1.34, enriched uranium: 1.79. Increases initially as Pu is produced from 238 U, decreases later as fission products are produced

About 0.9 for natural uranium. Larger as absorptions in nonfuel material decrease.

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

 About 1.05 for natural uranium.

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

L 2 Water: 0.027 m, heavy water: 1.0 m, graphite: 0.54 m

B 2 Typically less than 10 m -2 , therefore P NL1 > 0.97 and P NL2 > 0.99 for H 2 O

Time Dependence

• To study the time-dependence of the flux we have to solve the multigroup equations in slide 3 augmented to include the equations for delayed neutrons.

• There are two time scales:

 Short-term chan ges (seconds) due to tempera ture effects and external deliberate chan ges

 Long-term changes (hours or more) due to fuel deplet ion and fis s i on-product buildup.

Point Kinetics

• Recall (slide 6):

  1 ( r )  

  1   ( r )

 2  ( r )  B 2  ( r )  0 ,

 ( r s )  0

 

 2 ( r ) 

 

 2 

• Local perturbations, e.g., by moving the control rods, leads to a read justment of  ( r )

that is usually slight and happens in a fe w milliseconds. Th en, the readjusted shape rises or falls “as a wh ole” depending on whether the pertu r bation increas e d or decreased k.

• Point kinetics allows us to investigate th e level (or average) flux assuming that the shape d o es not ch ange ap preciably.

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

n ( r , t )  n ( t )  ( r )

• n(t) is the total neutron d e nsity or the total power

G

Power ( t )    dVw fg '  fg '  g ' ( r , t )  n ( t )

g '  1 V

• Using th is equation in the space- and time - d ependent equations and including delayed neutrons leads to

Point Kinetics Equations

dn ( t ) 

dt

 ( t )  



n ( t ) 

6

  i C i

1

( t )

dC i ( t ) dt

  i



n ( t ) 

 i C i ( t )

i  1 , 2 ,..., 6

  

k ( t )

  

1

v  a ( 1  L 2 B 2 )

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

Prompt neutron lifetime between birth of a neutron and absorption;



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

Delayed Neutron Precursors

Glasstone & Sesonske, Nucl ear Rea c tor En gineering, Chapm a n & Hall, 1994

Department of Nuclea r S c ien ce and Engineering 13

Reactivity Feedback

• 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 themselves involve the atomic number densities of the materials:

 ( r , t )  N ( r , t )   ( r , t )

• The atomic density depends on the power level because:

 Material densit ie s depend on temperature, which, in turn, de pends on the power dis tribut i on

 The buildup of poisons and burnup of fuel ( long-te rm effec t).

• We write

 ( t ) 

  ext ( t )    f [ P ]

 ext ( t )

External reactivity from some reference power level P 0 for

which  is zero.

 f [ P ]

Change in reactivity due to inherent feedback mechanisms.

Power Coefficient of Reactivity

d      

 T j  j   T j 

 P 

    

    T  

dP j

  T j 

 P  j

  P 

Safety requirement:

 P  0

  PD

FP

  dP  P 0

Powe r D e fect : Total reactivity change from hot zero-power state to hot full-power state; compensate d by control-r o d insertion and soluble boron. It is less than about 0.05.

Dominant Coefficients of Reactivity

   F   T F

   M   T M 

P T 





 P 

T   P 

 F 

   1

 k  1  k

T  T

k 2  T F

k  T F

Doppler broadening of resonance absorption decreases p.

For PWRs: (-4 to -1)x10 -5 Δρ /°K or (-4 to -1) pcm ( per cent mille)/°K .

Moderator: Thermal expansion leads to loss of neutr on moderation and a corresponding decrease in p. The decrea se in the density of poison atoms leads to reactivity increase. Overall e ffect should be negative.

For PWRs: (-50 to -8) pcm/ °K.

General Design Criteria 27 and 28

• Criterion 27--Combined reactivity control systems capability . The reactivity 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 the core is maintained.

• Criterion 28--Reactivity limits . The reactivity control systems shall be designed with appropriate limits on the potential am ount and rate of reactivity increase to assure that the effects of postulated reac tivity accidents can neither (1) result in damage to the reactor coolant pressure boundary greater than limited local yielding nor (2) sufficiently disturb the core, its s upport 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 addition.

Standard Review Plan: 4.3 Nuclear Design 2

The areas concerning reactivit y coefficients include:

The applicant's presentation of calculat ed nominal values for the reactivity coefficients such as the moderator co efficient, which involves primarily effects from density changes and takes the form of temperature, void, or density coefficients; the Doppler co efficient; and power c o efficients.

10 CFR 50.68 Criticality Accident Requirements (1)

• E ach licensee shall comply with th e following requirements in lieu of maintaining a monitoring system ca pable of detecting a criticality as described in 10 CFR 70.24:

 Plant procedures shall prohibit the handling and storage at any one time of more fuel assemblies than have been determined to be safely subcritical under the most adverse moderation conditions feasible by unborated water.

 The estimated ratio of neutron production to neutron absorption and leakage (k- effective) of the fresh fuel in the fresh fuel storage racks shall be calculated assuming the racks are loaded with fuel of the maximum fuel assembly reactivity and flooded with unborated water and must not exceed 0.95, at a 95 percent probability, 95 percent confidence level. This evaluation need not be performed if administrative controls and/or design features prevent such flooding or if fresh fuel storage racks are not used.

10 CFR 50.68 Criticality Accident Requirements (2)

• If no credit for soluble bor on is taken, the k- e ffective of the spent fuel storage racks loaded with fuel of the maximum fuel assembly r eactivity must not exceed 0.95, at a 95 percent probability, 95 percent confidence level, if floo ded with unborated w ater. If credit is taken f o r soluble boron, the k-effective of the spent fuel storage racks loaded with fuel of the maximum fuel assembly reactivity must not exceed 0.95 , at a 95 percent probability, 95 percent confidence level, if flooded with borated water, and the k-effective must remain below 1.0 (subcritical), at a 95 percent probability, 95 percent confidence level, if flooded with unborated w ater.

• R adiation monitors are provided in storage and associated handling areas when fuel is present to detect excessive radiation levels and to initia t e appropriate safety actions.

• T he maximum nominal U-235 enrichment of the fre sh fuel assemblies is limited to five (5.0) percent by weight.

Core Composition Changes – F ission Product Poisoning

• Some fission products have large thermal absorption cross section. The poisoning effect is insignificant for fast reactors.

• Most important products:

135

54

with

 X  3 x 10 6 b  3 x 10  22 m  2

and

149

62

with

S  5 x 10 4

b  3 x 10

 24

m  2

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

causes.

    '   

k '  1 

k '

k  1 k

 1  1 



k 

k ' 

  F 

• The thermal utilization

f   a 2 

is the only factor that is appreciably

affected by the poison.

  a 2 

  

1  1 

k  

 1  f

  1 

 a   aP

   aP

 k '  

f ' 

 a  a

Xenon Poisoning

γ I = 0.061 γ X = 0.003

λ I = 2.9x10 -5 s -1

λ X = 2.1x10 -5 s -1

X  3 . 0 x 10  22 m  2

 I ( r , t )

 t

  I  f  ( r , t ) 

 I I ( r , t )

 X ( r , t )

 t

  X  f  ( r , t ) 

 I I ( r , t ) 

 X X ( r , t ) 

 X  ( r , t ) X ( r , t )

I  

 I  f  0

X  

(  I

  X )  f  0

  

  X (  I

 I

  X )  f  0

 X

   

  X  0

17 2

a

(  X

  X  0 )

0 . 023 for

 0  10 n / m s

 I ( r , t ) 

 t

  I I ( r ,

Xenon and Reactor Shutdown

t )

 X ( r , t )

 t

  I I ( r , t ) 

 X X

( r , t )

A reactor operating at a flux of 2x10 18 n/m 2 s will have a negative insertion of reactivity of about -0.33, a sizable amount.

Glasstone & Sesonske, Nucl ear Rea c tor En gineering, Chapm a n & Hall, 1994

Fuel Depletion (Burnup)

• Du ring reactor operation, fuel ( 235 U) is depleted and new fuel ( 239 Pu) is produced.

• There is a net decrease of reactivity over time.

• In general,

   a t  

   a  

N F ( r , t )

N F ( r , 0 ) ex p   F   ( r , s ) d s 

N F ( r , 0 ) exp F ( r , t )

 0 

 ( r , t ) 

t

  ( r , s ) ds

0

is the neutron fluence

• The dependence of the flux on the fuel density complicates the calculations.

• In lieu of the fluence, it is customary to use the thermal energy outpu t per unit mass of fuel (burnup) in MWD/T (MW days per metric ton of uranium fue l) .

• Cu rrent limit: 62 GWd/T. F o r a 1,000 M W PW R, burnup is about 15 GWd / T after one year.

REGULATORY GUIDE 1.77

ASSUMPTIONS USED FOR EVALUATING A CONTROL ROD EJECTION ACCIDENT FOR PRESSURIZED WATER REACTORS

• “In general, failure consequences for U0 2 have been insign ificant below 300 cal/g for both irradiated and unirradiated fuel rods. Therefore, a calcu l ated radial average energy density of 280 cal/g at any axial fuel location in any fuel rod as a result of a postulated rod ejection accident provides a conservative m a ximum lim i t to ensure that core damage will be min i m a l and that both shor t-term and, long-term core cooling capability will not be impaired.”