Lý thuyết Lò phản ứng

- important to avoid un-damped oscillation Large flux distortion (Axial offset, Axial Shape Index) leads to safety problem, such as CHF damage of fuel rod cladding, fuel melting during[r]

3 Reactor transients

Separation of variables

neutron lifetime

infinite reactor lifetime :

nl

P 1Pnl

number of neutrons at time k

number of neutrons at time eff 



1

eff eff

k N N k

dN N dt      

N : number of neutrons

vN

 thermal reactor : v is small, ais large l~ 10-4~ 10-3s

fast reactor : v is very large, ais small l~ 10-7~ 10-6s

example:

k=1.001, l=10-4s 10 ( ) / t

N t N e e10 ~ 22, 000

impossible to control

fission bomb : l~ 0.3x10-8 k~1.4 doubling time ~ 5ns 

Reactor kinetics

 

,

v f a

D r t

t



     

     



 r t,    t r

  

   

2

0

r B r

 

  

     

1

v f a

d

t t DB t dt      

 2

1

va B L Pnl

 



 

v v a a                    2

v /

1

v 1

a f a

eff a eff d DB t dt k

DB k t t

precursor equations

Transient diffusion equation

separating space dependency

v

n neutron generation time v f

=

k v

  

1 k

k

   Point kinetics equation

Reactor kinetics with delayed neutrons

2

     

1

, ,

v a f i i i

D r t C r t S

t

       



        

 

 ,  , for 1, ,

i

i f i i

C

r t C r t i t   

   

 

1     

1

v v i i i

k d

t C t S dt



    

  



  i i 

i

dn

n t C t S

dt             i i i i dC

n t C t

dt      reactivity Laplace transformation i i i            i i i           InHour equation

 

0

i

t t

i i

1, 2, … <

0 when

  

0 when

  

asymptotic behaviour ( t >> 0)

  0t

n t e

stable reactor period

0 T    $     0.5$ Asymptotic behaviour when >> 0

    

 T  

 

 when<<

i i i       1      



 

  



T  

  

when<<

0 i i i                                Inhour equation

~

short term long term

 

t

n t n e

  

    



prompt jump

prompt jump

when < 

Prompt jump

4

 

t t

n t n e e

  

 

 

   



 

 

 

   

 

 

 

when>>

Prompt critical excursion

when > 

 

t

n t n e

 

  

 

 

avoid reactivity insertion more than 1$

prompt drop For large negative reactivity insertion

(trip: control rod drop)

   1/ 2,1

1 55

80( )

ln 0.693

T

T s



      

when  < 0

Reactor trip

Br-87

half life : 55.6 sec fission yield :

conversion ratio

9

9

9 9 ,

U U

U

U U a U n U

dN

N N N

dt       

9

9 9 9

Np Np

Np Np U U a Np

dN

N N N

dt     

9

9 9 9

Pu Pu

Pu Pu Np Np a Pu

dN

N N N dt     

conversion

5

5 5

U U

U U a U

dN

N N

dt    

burnup

 -1

day

 a( )b n,  b

half life

 24 2  14   

680 10 10 / / s 24 3600 /

a cm cm s d

       

= 5.9x10-3/d

8

8 8

U U

U U a U

dN

N N dt    

8 2.3 10 U a      8 U U a U dN N

dt   

simplified 3.1 10 U a     3.1 10 U a     

9 , U U

U U n U

dN

N N

dt     

cannot be ignored for balance 5.9 10 Np a    

9 9 Np

Np Np U U

dN

N N

dt   

9 8.8 10 Pu a      9

9 9

Pu Pu

Np Np a Pu

dN

N N dt   

Burnup and conversion

Delayed conversion

 

232 233 233 233

22.3m 27.0d 159000

Th n Th  Pa  U a

238 239 239 239

23.5m 2.36d (24110 )

U n U   Np  Pu a

reactivity increase after few days (or few weeks)

Isotopic depletion

8

Xe-135 and Sm-149

2600000b



   40000b

Important in reactor control

simplified model

2600000b



   8.3b  6b

0.01b



 

135 135 135

6.6h 9.2h

I Xe Cs

I

 X

equilibrium xenon

reactivity change due to xenon

Xenon poisoning Delft HER reactor

=0.037

2.3

 

 need excess reactivity to compensate xenon poisoning

Equilibrium xenon and iodine

after reactor shutdown (flux=0)

I

dI

I dt  

X I

dX

X I

dt   

 

It

I t e I

when xenon increase

 

X I X f I f X aX              

11 10 X I I aX cm s           transient

0 0

Xt I I It

X I X I

X e  X  I  e  I

   

   

   

 

 

peak time dX dt    ln / 11.2 I X I X

t   h

 

 



xenon dead time Xenon dead time

10

 

0

I X f X aX

X   

  

 



 I0  I f / I

0

Xenon oscillation coupled with spatial flux distribution

xenon build up k∞ decrease control rod

xenon decays k∞ increase

time

~20 hours

flux decrease

flux increase

flux increase

ref J.S Song et al 1999

- check xenon stabilty

- important to avoid un-damped oscillation Large flux distortion (Axial offset, Axial Shape Index) leads to safety problem, such as CHF damage of fuel rod cladding, fuel melting during accident

Xenon oscillation

12

control rod insertion

excess (over) reactivity is required for reactor operation to compensate - reactivity decrease by temperature (power) increase

- burnup (depletion) of fuel and fission product build up - xenon and samarium poisoning

- control reactor

keff

xenon

fuel depletion - (solid) burnable poison (boron,

gadolinium)

- soluble boron (H3BO3) : speed 500l/min (full replacement ~ 15hr) - control rod : speed 30’’/min (full

insertion ~ 5min)

Soluble boron depletion

Soluble boron requirement

BOC EOC

Boron worth : ~ -7pcm/ppm Excess reactivity control

Control rod overlapping

top top

differential worth accumulated worth

Control rod worth

 

2

d

16

Summary

 

1

,

v t D r t f a

     

      



speed of neutron : ~2,200 m/s Reactor size : ~ 1m

Diffusion coefficient : ~ 0.1 cm

2

1 / v

D t

R

  ~ 0.5 sec

We may assume transient become equillibrium in few seconds Time constant of system

• Dealyed neutron kinetics

- solve time dependent diffusion equation with initial value - Point kinetics is useful for trip problem

- Diffusion equation is required for rod ejection problem • Xenon transient, Depletion time constants ~ hours