Chapter 5 Kinetics

Fission chain r eaction

Public domain image from wikipedia.

Delayed neutrons

Delayed neutrons emitted from the decay of fission pro d uc t s l ong a ft er th e fi ss i on even t . D e l ay i s cause d b y half-life of beta decay of delayed neutron pre-cursor nucleus .

Delayed neutrons

Delayed neutrons are grouped into 6 groups of delayed neutron precursors with an average decay constant  i d e f i n e d for each .

 i  T he dec a y co nsta nt fo r the i th g r o u p o f de lay ed ne utro ns

 i  T h e f r a c tio n o f f i ssio n ne utr o ns e m itte d by de la y e d ne utro n pre c u r s o r g r o u p i.

  T h e f r a c tio n o f f i ssio n ne utr o ns tha t a r e de lay e d.

6

=   i i=1

 ~ 0 . 0 0 7 5

Point Kinetic Equations

For an initially critical system

dp ( t ) 

  ( t )  

 p ( t )  K

c ( t )

d t   

 k k

  k  1

dc k

( t )  k

p ( t )   c

( t )

d t  k k

Most important assumption: Assumes that the perturbation introduced in the reactor af fects only the amp litude of the flux and not its shape.

Dynamic Reactivity ρ (t)

• Most important kinetics parameter

– Its variations are usually the source of changes in neutronic power

– O nly term that contains the neutron loss operator (M operator)

• A ssociated to control mechanisms

• A lso sensitive to temperature

– N o units

• E xpressed in terms of mk (milli-k) or pcm

Delayed - Neutron Fraction β (t)

• E ffective dela y ed neutron fraction is linked to the constants of each fissionable isotope which measure the fraction of fission product precursors

– C alled “effective” because it is weighted by the flux in the reactor

• C an vary with burnup

– Different values exist at BOL and EOL

– Variations in burnup are on a much larger time scale than usual range of application of point kinetics equations

Prompt - Neutron Lifetime

• Measure of the average time a neutron survives after it appears as either a prompt neutron or a delayed - neutron

Solution with one ef fective delayed neutron precursor group

dP ( t )

   0  

 P ( t )   C ( t )

dt     

d C ( t ) dt

 



P ( t )   C ( t ) , t  0

Solution

P ( 0 ) 

P , C ( 0 )

  P

  0

Initial conditions

P ( t )  Pe st , C ( t )  Ce st

Solutions

sP    0  

 

 P   C



Substitute

s C   P



  C

 s    0

  

Solution

  

  

  

  P   0  Linear

   C    0  Homogeneous

  

s    

   System

    

 s    0

     s    

  0

      

Non-trivial solution if and only if

 s 2  (       0 ) s   0   0

det A=0

Solution

s  1            

1 , 2

2   

0  

P ( t ) 

Pe s 1 t 

P e s 2 t

and

C ( t ) 

C e s 1 t

 C e s 2 t

1 2 1 2

Approximate solution

Assume

  / 

 1

s   0

, s    

  0 

 0  

   0

    

P ( t )  P

   

exp

  0

 t    0

 exp

    0  t 

0     

     

    

  

  

  0   0   0  

Solution

This solution is not valid for large changes in reactivity!

P ( t )  P

        0  t

  0

   0

   

0      

  ex p    

      

  ex p 

   t 

  0  

0  

0  

 

 0 0.002 5 ,  0.007 5 ,  0.0 8 s -1 ,  10  3 s

s  1 ~ 25 s

1

s  1 ~ 0 . 2 s

2

Prom p t neutrons

Delayed neutrons

Reactor Period

• D efined as the p ower level divided b y the rate change of power

( t ) 

p ( t )

– Period of infinity implies steady-state

– S mall positive period means a rapid increase in p owe r

– S mall negative period means rapid decrease in power

– If period is constant, power varies according to

p ( t )  p 0 e t / 

Reactor Period

• For the case with one delayed group , the

reactor period can be separated in two parts

– P rompt period

– Stable period

• T he solution has two exponential and they usua ll y h ave very diff eren t coe ffi c i en t s.

• ρ 0 < 0

Scenario 1

– C orresponds to a quick reactor shutdown

– Both roots are negative

– s 2 <<< s1 thus the power drops almost instantly to a fraction of its initial power (prompt drop)

– However , it is impossible to stop a reactor

instantaneously

Example

• T h e s e c o n d r o o t is s o

• T h u s

the s t a b l e

small that in the matter of a fraction of second becomes inconsequential

• P ower drops almost instantly to the coefficient of the f i r s t exponential term

period is equal to 1/s 1

• A nd the p rom p t period is equal to 1/s 2

1

0. 9

0. 8

0. 7

0. 6

0. 5

0. 4

0. 3

0. 2

0. 1

0

0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1

P ( t )  P

    e x p    0

 t    0

 e x p   0    t 

0      

    

    

   

  0  

0  

0    

Courtesy of canteach@candu.org. Used with permission.

Demo

• N e g a t i v e S t e p i n s e r t i o n o f - 12mk

• P arameters

– B e t a = 0 . 006

– L AMBDA = 0.001

– Lambda = 0.1 s - 1

• P ower d rops b y 3 3 % a l mos t i ns t an t l y, an d then decays slowly

• 0 < ρ 0 < β

Scenario 2

• O ne root is positive and the other is negative

• P ower increases rapidly and the grows exponen ti a ll y

• Power increases rapidly by beta/(beta - rho)

– P ositive prompt jump

• Stable period is equal to 1/s 1

• P rompt period is equal to 1/s 2

1. 4

1. 2

1

0. 8

0. 6

0. 4

0. 2

0

0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1

Demo

• Positive Step insertion of 1mk

• P arameters

– B e t a = 0 . 006

– L AMBDA = 0.001

– Lambda = 0.1 s - 1

• If rho approaches beta, the stable period becomes very short

• ρ 0

Scenario 3

> β (prompt s uper - critical)

• R eactor is critical without the need of the delayed neutrons

• O ne root is positive and one is negative

• R eactor period becomes less than 1s

• Power increases at a very rapid rate

• D isastrous consequences

– U nless a feedback mechanism can cancel out the reactivity

8

7

6

5

4

3

2

1

0 0 0. 01 0. 02 0. 0 3 0. 04 0. 05 0. 0 6 0. 07 0. 08 0. 09 0. 1

Demo

• Positive Step insertion of 7mk

• P arameters

– B e t a = 0 . 006

– L AMBDA = 0.001

– Lambda = 0.1 s - 1

• R eactor is critical (or supercritical) without the presence of delayed neutrons

– P rompt jump dominates

Limiting cases - Small reactivity insertions

   , thus s     ... 

l  1

T  1  1  l

  6  i   

s  

i  1   

k  1

l

l

1 0  i  0

   , thus s  

0 1 i

Limiting cases- Lar ge reactivity insertions

  s 1

 l  1

   s 1   l  1

s 1  l  1

s 1  l  1

i

i  1

s 1  l  1

T  1  l  l s 1 k (  0   ) k  1

Positive Reactivity

Negative Reactivity

Typical parameters

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22.05 Neutron Science and Reactor Physics

Fall 20 09

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