Lý thuyết Lò phản ứng

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(1)<div class='page_container' data-page=1>

2. Neutron interaction and

transport

</div>
(2)<div class='page_container' data-page=2>

Neutron

interaction

(n,3n)
elastic scattering

capture (n,)

fission (n,f)

(n,n’)

235

U

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(3)<div class='page_container' data-page=3>

1 1 *
0

A A

Z Z

n



X





X

capture

fission

absorption
Reaction : Compound nucleus

 





reaction channel

Elastic Scattering kineticsenergy conservation
momentum conservation

recoil

scattered neutron

(n,n) – elastic scattering

scattering

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(4)<div class='page_container' data-page=4>

Cross section (microscopic) :
probability of a reaction channel

unit : barn (area) = 10-24cm2=100fm

 

number of ptl. scattered into solid angle

per unit time

incident intensity

d



  



2 sin

d

 



 

d

s

 

ds

d





number of interaction per unit time per unit area
incident intensity (number per unit time per unit area)

a

N





for very thin film with areal atom density Na

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(5)<div class='page_container' data-page=5>

a

dI



N I

N Idx



 



 

N x

I x



I e

 

nuclide density

Macroscopic cross section

N



 

unit : cm-1

 

x

I x



I e



Mean free path
0

x

xe dx



   





Number density

A

N N

A








(gr/cm )3





barn 0.6022 cm

(u)
A





    

Macroscopic

cross

section

</div>
(6)<div class='page_container' data-page=6>

fission

moderation

absorption

moderation leak

Speed of neutron : 1 2

v
2
E m

neutron speed time to travel 1m
1MeV:13,800 km/s 72.3 ns

1eV:13.8 km/s 72.3 ms

25.3meV : 2200m/s 0.454 ms

v 2 /E m

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Monte Carlo simulation method

- Tracking individual neutron flight and collision history
- statistical average behaviour

scattering
born

fission

capture

tot



/ tot
p  

n n tot

p  
/

f f tot

p  

free

tot
l

N



Monte Carlo simulation

Neutron

transport

Monte Carlo method computer codes for neutron transport
(and eigenvalue problem)

MCNP : developed by LANL, USA
McCARD: developed by SNU, Korea
SERPENT-2 : VTT, Finland

KENO (SCALE package) : developed by ORNL , USA
• can describe physics accurately

• easy to handle complex geometry

• good to know a value at a detector volume
- weight biasing to improve statistics
• not good for sensitivity study

• requires long computer time for good statistical error
- parallel computing using MPI

- vector computing using GPU
- precomputed MC

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(8)<div class='page_container' data-page=8>

Reaction rate : R



r, ,E   t





r, , ,E t

 

 r, , ,E  t



(m-3·rad-1·s-1)

Number of neutrons in a control volume:

Neutron balance in volume V, energy interval dE, angle interval d

Neutron

transport

equation

‐

derivation





, , ,

Vn E  t dr dEd



r

1. neutron source





, , ,

VS E  t d rdEd



r

2. scattering of neutrons from other energies and directions



 



0 4 s , ' , ' , ', ', ' '

V  E E  E t dE d dr dEd



       

  

r r

3. net outflow of neutrons









ˆ , , , , , ,

Sn E  t dEd dS   V  E  t dr dEd



r



r

Gauss’s
theorem
4. neutrons interaction



 



, , , ,

t

V E  E  t dr dEd



r r





, , ,

n r E  t d dEdr 

Number of neutrons at time t in volume dV, energy between E and E+dE,
moving toward solid angle between  and +d

(m-2·rad-1·s-1)

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- linear integro-differential equation

- Boltzmann transport equation ignoring (n-n) collision term

Initial condition and Boundary condition on convex volume



, , , 0E



 r  



rs, , , 0E 



0 for  nˆ<0

no incoming neutron flux

non-convex volume can be treated
by larger convex volume, always

Neutron

balance

equation











 









 



3 3 3

0 4

3 3

, , , , , , , ' , ' , , ', ', ' '

, , , , , , ,

s

V V V

t

V V

n E t dr dEd S E t dr dEd E E t E t dE d dr dEd

t

E t dr dEd E E t dr dEd

 
 


              

        



  



r r r r

r r r

Recall definition of flux  vn

Time dependent neutron transport equation

Chain reaction problem 1 f ext
eff

S S

k  

  

Eigenvalue problem







 



0 4



 







, , , t , , , , s , ' , ' , , ', ', ' ' f , , ,

eff

E t E E t E E t E t dE d E t

k



     

  r    r r  

 

 r     r     r 











 





 







0 4
, , ,
1
, , , , , , ,
v
, ' , ' , , ', ', ' ' , , ,
t
s
E t

E t E E t

t

E E t E t d dE S E t



 


 
     



 

        
r

r r r

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(10)<div class='page_container' data-page=10>

Solving

transport

equation

Difficulties

- Energy dependent cross section is widely varying

- resonance cross section is dependent on temperature

 Multigroup condensation
- Angle dependency

- Peripheral is strong, Center is weak

 SN, PN, Diffusion, etc.
- Complex geometry

- lattice structure

 Homogenization

- Numerical difficulty in convergence and negative flux

 Diffusion approximation

sufficient to predict power distribution in LWR



, , ,E t



 r 

Find the multiplication factor (eigenvalue) and the flux (eigenvector)

reaction rate : 
power : f 

eff

k

neutron streaming/
vessel fluence

homogenization

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(11)<div class='page_container' data-page=11>

Slowing

down

Neutron slowing down – mostly by elastic scattering





2
2

2
'2

2 1

' 1

c

A A

E v

E v A



 

 



Energy after elastic scattering Maximum energy transfer

Average logarithmic energy decrement

Average number of collisions





1 1

1 ln

2 1

A A

    



2
2 / 3

A
 









E E'/



N E E



 

</div>
(12)<div class='page_container' data-page=12>

Moderation

 

lnE0

u E

E



practical highest energy : E0= 20 MeV
Good moderator material

• large s

• large 

• small a

Moderation data for some element
Lethargy

slowing down power



s average logarithmic energy loss per unit path

s a



 

moderation quality

enriched uranium is needed
natural uranium can be used
thick reflector is needed. ~ 1meter.

(reactor is bulky)

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Assumptions :

• above thermal energy (> 1 eV), nuclides is considered as not moving before collision.
• ignore absorption. (reasonable at moderator region due to 1/v behavior)

for single isotope

 

s

   

F E   E  E

let F E

 

C

E



   



 

  

' ' 0

' 1

E
s
s

E

E E E dE

E



 





  





Slowing

down

density

   

0



  

' ' 0

s E  E s E E  E dE



  



  

Infinite homogeneous medium

Slowing down density : number of neutrons that passes the Energy E per volume per time





' '' ' ' '' '' '

E E
s
E E E E

q  E E dE dE C

 

 



 

  

slowing down density is
independent of energy

(if no absorption)

 

 

s
q
E

E E





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(14)<div class='page_container' data-page=14>

Epithermal

spectrum

el



is nearly constant (except near resonance)

u

EE e

 

u
u

s s

qE e du q

u du E dE du

E e

 

 




    

 

flux is constant in lethary scale

 

s

q
E

E







</div>
(15)<div class='page_container' data-page=15>

 





3/ 2 1/ 2 /

2 E k TB

B

N E dE

E e dE

N



k T





 

2 

3/ 2

/ 2
2

v v

4 v v

B

mv k T
B

N d m

e d

N k T 



 
  
 
2
1
v
2
E m

speed distribution

Thermal

energy

Thermal equilibrium of atoms follows Maxwell Boltzman distribution

Neutron scattering with medium in thermal motion









2
/ 2
'
', , , ,
' 4
b
d E

E E T e S T

d dE kT E



   
 
 

b

 bound atom scattering cross secion





2 2
4
1
, ,
4

S T e

 

 





Free gas model

E E

kT

  

' 2 '

E E EE

A kT


   

energy transfer
momentum transfer

A0 : Mass number of atom

E

E kT

A0

cos 

• neutron can gain energy from
moving target

 Scattering matrix, S(,), is depending on
molecular structure, and crystal structure
(related to photon dispersion relation)

</div>
(16)<div class='page_container' data-page=16>

Scattering

in

graphite

Ref) INDC-NDS-0475 (IAEA 2005)

free gas model

Bragg’s cutoff

</div>
(17)<div class='page_container' data-page=17>

 





1/2 /
0 3/ 2

B

E k T

B

N E dE N E e dE

k T







neutron flux

 





03/ 2 /

2 2 E k TB
n

B
N

E dE Ee dE

m
k T










most probable energy
=kBT

At T= 293K
kBT = 0.025 eV

= 2200 m/s
At T= 590K
kBT = 0.051 eV

= 3100 m/s

 neutron temperature is higher than
medium temperature.

low energy cross section is higher

abs ~ 1/v

Thermal

neutron

flux

distribution

</div>
(18)<div class='page_container' data-page=18>

  







0
,
0
,
,

a M n

a th

M n

E E T dE
E T dE

 








 





/
0
3/ 2

2 2 E k TB

n

B
N

E dE Ee dE

m
k T






 

 



2
/

0 2

B

E k T

B B

Ee dE k T k T

 
  



0
293 B
a a
k
E
 





1/ 2
, 0
0
293
2
293
2

a th a B B

a

k k T

T

 
 












/ /
0 0
3/ 2
1

3 / 2

B B

E k T E k T

B

Ee dE Ee dE

E
k T

   
  




Effective

absorption

cross

section

consider 1/v behavior of absorption cross section

a

 cross section at 2200 m/s

Maxwell average cross section at temperature T

a th



Recall gamma function

</div>
(19)<div class='page_container' data-page=19>

Heavy nuclides

• resonance energy region : 1 ~ 100keV
• interval is small : ~ 10 eB

• energy width is narrow
• amplitude is strong
Light nuclides

• resonance appears at very high energy : ~MeV
• energy width is wide

• amplitude is not strong

</div>
(20)<div class='page_container' data-page=20>

interference
upper

bound

Breit-Wigner single level resonance formular



 



2 2

4 / 2

n a
na

r

E E






 


  

reaction channel (,f, etc.)

2 n n

h h

p m E

  

de Broglie wave length of neutron

For low energy E~0

1 1

na

E

E E

   

4 potential scattering term

s a

   

n E

 

2 1

lJ

n J nn

l J

g U

k


 





/ 2

i nr

nn

U e i

E E i



  

 

 

  



elastic scattering

penetrability factor

collision matrix

ka
 

For low energy

1/v law of
neutron cross section

~ constant.

a



Resonance

cross

section

</div>
(21)<div class='page_container' data-page=21>

238U

 

2 0

0 4 0 2 0

n

E g  



      

 


at resonance peak

 

2
6
2
0
0

2.608 10 1

4 b
eV
A
E A
     
 


 





2
2 2
0
1
4
n
E g
E E


   

  


near resonance peak

 

 

 





2
0
2
0 0
0
2
1
4
1
l
l
P E
E
P E
E E


    
  
     

 



 





2 1 / 2
0
2
0
0
2
1
4
1
l
E
E
E
E E


 

  
  
      
  
 

(exact for s-wave)

 

0 1/ 2





0 2
1
1
E
E
E y


    
   

for s-wave resonance





y EE



Resonance tail (E ~ 0, s-wave)

1/ 2
0

0 2 2

1 1

1 4 /

E

E E E




     

    

</div>
(22)<div class='page_container' data-page=22>

Average absorption in resonance energy range

• (broad) flux is 1/E outside narrow region of resonance
(constant in lethargy unit)

   

 

a
resonance
res
resonance

E E dE
E dE

 






,
,
a res
a res
res
RI
u
 


Resonance Integral a res, a

 

a

 

resonance resonance

dE

RI E u du

E

 









Resonance

Integral

 

0 1/ 2





0 2
1
1
E
E
E y


    
   

 

res
E
RI dE
E






integrate only y, high value near resonance and full integration is ∞





/ 2

2 2
2 / 2

1 1 1

1 tan cos

1 y dy d



    

 

     







y EE



2
dEdy

0 2

1 1

2res1

RI dy
E y



 






0
0
2
RI
E

 




</div>
(23)<div class='page_container' data-page=23>

Target nucleus are moving due to temperature

relative velocity of neutron with moving nuclide vr  v V v

V

 

p V dV : probability of a nucleus having velocitity between (V, V+dV)

effective cross section of neutron for target temperature T

 

   

1
v
v

T E r Er P d

 

 



 V V

probability of reaction per sec vr

   

Er P V dV

relative energy ?









2 2 2

v v

2 2

r z x y

m m

E  V   V V V  (assume neutron is moving z direction)





2
2

r z

m

E  v  vV (neutron is much faster than target velocity)

Doppler

broadening

 

2 

3/ 2
/ 2
2
4
2
B

MV k T
B

M

P d V e d

k T 





 

  

 

V V V

assume Maxwell distribution of target nucleus

 

,T 0 ,

E
E x
E


    

 Doppler line shape function

 





1/ 2
0
0 2
1
1
r
r
E
E
E y


    
   
s-wave SLBW
We have

 

,x

</div>
(24)<div class='page_container' data-page=24>

When T is low, is large

 

1
,
1
x
x

  


When T is high, is small

 

1 2 2

, exp

2 4

x  x

      

 

Line shape function for scattering cross section

 





2
2

exp / 4

1
2

y x y

x dy
y


 



 





Doppler

line

shape

function

U-238 0K
300,000K

 





2
2
2
1
exp
4
,

1
2
x y
x dy
y


 



  
 
 





Line shape function for absorption cross section





r

y E E







x EE



4k TEB 4k TEB r

A A

  

  

</div>
(25)<div class='page_container' data-page=25>

Absorption in narrow energy range

homogeneous mixture with fuel, ignoring absorption in moderator

• scattering cross section is nearly constant

• (broad) flux is 1/E outside resonance (constant in lethargy unit)

Homogeneous

mixture

   



  

' ' ' 0

E

t E E E s E E E dE



   

 



 





 



 





 



' '

' 1 ' 1

F M

E E

s s

F F M

a s s

E E

F M

E E

E dE dE

E E

     

   

 

   

 



 

 

1 /

u
a

E

E E




 




What is flux shape at resonance ?

pot



a



E

resonance

(U-238)

</div>
(26)<div class='page_container' data-page=26>

Scattering resonance at fuel(resonance) nuclide is small

 



 



 

' '
1 '
R
E s

R R R b

a s pot b E

E

E E E E dE

E E
  
     

      
 





• NR (Narrow Resonance) approximation

 assume resonance width is small

 





 

/ / 1

' ' '

1 ' 1 ' '

R R

R

E pot E pot

s u

u

E E

E

E dE dE

E E E E

     

 

   

 



 

 

Rpot b 1

R
t b
E
E E
 

 




   





 

aR

 

R R

NR a pot b R

t b

res res

E

RI E E dE dE

E


   
 
  




total

b: background cross section (include all others)

Large background = infinite dilution

 

lim

b b
R
a
NR WR
res
E

RI RI dE

E

 



 

 



Resonance

approximation

• NRIM (Narrow Resonance Infinite Mass absorber) or Wide Resonance approximation

 integral term is zero

Infinite dilution

 

 

b 1

R
t b
E
E E


 



 

R
a
WR b R

t b
res
E

RI dE
E


 





</div>
(27)<div class='page_container' data-page=27>

Discrete

Ordinate

method

(S

N

method)

Neutron transport equation (steady state)

Angular quadrature









2 1

' '

s s P

 






   



   




Scattering cross section (in Lab system)



, ,E



t



,E

 

, ,E



0 4 s



,E' E, '

 

,E', '



dE d' ' S



, ,E





     

  r   r r  

 

r     r    r 

 

N

n n
n

fd w f





 







n : ordinate

wn =n : angular weight
scattering integral term becomes summation

transport collision scattering source

n





N

n n t n j s n j j n
j

w S

    



   



  

Solve NxN system of equation for angular direction
Spatial discretization : FDM, etc.

ANISN, DORT, TORT (ORNL), DANTSYS (LANL), etc.

</div>
(28)<div class='page_container' data-page=28>

Spherical

harmonics

(P

N

)

method

Angular flux is approximated by a finite spherical harmonics expansion





   

0
,
N n
m m
n n
n m n

r r Y

 

 

 

 

 Total (N+1)2terms

Simplified PN method

1D case using orthogonality of Legendre polynomial

1 1

0 0

2 1 2 1

n n

n n n

d d

n n

q

n dx n dx

      

   

  isotropic source term

1 1

2 1 2 1

n n n n

d n n

dx n n

   
 

 
    
 
 

Elliminate odd order terms

1 1

1 2 1 2 0 0

1 1 1 2

2 1 n 2 1 n 2 1 n 2 1 n 2 3 n 2 3 n n n n

d n d n n n d n n

q

dx n  dx n  n  n  dx n  n    

 
   
        
        
         
 

0, 2, 4,

n 

for

Total (N+1)/2 equations
3D case

1 1

1 2 1 2 0 0

1 1 1 2

2 1 n 2 1 n 2 1 n 2 1 n 2 3 n 2 3 n n n n

n n n n n n

q

n  n  n  n  n  n    

 
   
        
        
         
 

SP7 have 4 equations

   

1
1

sn Pn s d

    






0 t s0 a

   

for 0

n sn n

</div>
(29)<div class='page_container' data-page=29>

Collision

Probability

Method

(CPM)

Assume isotropic angular flux and source



, ,E



t



,E

 

, ,E



0 4 s



,E' E, '

 

,E', '



dE d' ' S



, ,E





     

  r   r r  

 

r     r    r 

 

   

 

4



 r d t r  r q r

 

3
2

ˆ ' '

4 '

t

e

q d







 







r r

r r r

r r

CPM

 

   

k

n

t i i V s k k j k j ij i j j
k

r V r S r P r r dV

   



 





discrete volume

Pij : probability for a neutron starting at zone i colliding at zone j

i
j

Pij can be derived analytically (for simple geometry)
Widely used for multigroup condensation

'
3
2

4 '

t

ij

e

P d





 







r r r

r r

</div>
(30)<div class='page_container' data-page=30>

Method

of

characteristics



, ,E



t



,E

 

, ,E



0 4 s



,E' E, '

 

,E', '



dE d' ' S



, ,E





     

  r   r r  

 

r     r    r 





t

  



Q





  

 r   r r   r 

d
ds




 

along a characteristic direction

 

0 0 '

ts ts s ts

s e  e  Qe ds

    



analytic solution exists

DeCART : developed by SNU, Korea
OpenMOC

Ray-tracing method

</div>
(31)<div class='page_container' data-page=31>

Diffusion

equation

SP1 form

1 0 0 0 0

3    q



    

Fick’s law

J   D 

J  q

   

Diffusion equation

1
3

D





0 : absorption cross section

1 : transport cross section

 

1
1

tr s d

   









1 0



tr s

  

0 : average cosine angle of scattering

tr: transport mean free path

Finite difference method : need small intervals (1~2cm)
CITATION, VENTURE, PDQ, …

Nodal methods : large intervals (10~20cm)

ANM (MIT) : Analytic Nodal Method – 1D analytic solution with quadratic poly.
interpolation in transverse direction leakage

QUABOX, CUBOX (KWU) : polynomial expansions
MASTER (KAERI) : Nodal expansion method

</div>
(32)<div class='page_container' data-page=32>

Solution

Methods

for

Neutron

Diffusion

Equation

• Finite Difference Method

– Divides system into fine meshes equivalent to thermal neutron mean free

path (1~2 cm)

– Approximates flat flux in each homogeneous mesh

• Nodal Method

– Nodes are relatively large (10 ~ 20 cm) homogeneous volumes so that

FDM is not applicable.

– In the nodal equation, each node is only coupled with its direct neighbors.

– The average flux per node and the net leakage should be uniquely

</div>
(33)<div class='page_container' data-page=33>

Nodal

Method

for

Neutron

Diffusion

Equation

• Diffusion Equation Formulation

– Multi-group diffusion equation for steady state condition for flux and

current

∙

Σ

→

</div>
(34)<div class='page_container' data-page=34>

Nodal

methods



Assume intra flux distribution in homogeneous volume using various values

Partial current : net current = in coming – out going

xr xr xr

J J J

xr
J
xr
J
xl
J
xl
J
Surface flux
xl

xr

x

h



 





 



1 1

xr xr xl xl yr yr yl yl

x y

J J J J J J J J q

h h 

       
          
   
x
L

 

x
xs xs
x s
d x

J J D

dx


 



   sl r,

 

 

2
2
x
x x
d x

D x q L x

dx


   





xs Jxs Jxs

    

Partial current 1

2
x
x x
d

J D
dx


  
   
Transverse Leakage

 

,
x

y

D

L x x y dy

h y


 



</div>
(35)<div class='page_container' data-page=35>

Nodal

Method

for

Neutron

Diffusion

Equation

 

x xn n

n

x a p x

 



x

 

xn n

 

n

L x 



b p x

 

1 for 0

0 for 0

x

h
n
x

n
p x dx

n

h




  




Nodal Expansion Method (NEM)

- polynomial expansion of transverse surface flux and leakage

coefficients a and b can determined from nodal balance equations
Analytic Nodal Method (ANM)

- solve transversed integrated equation analytically

- assuming quadratic shape for the transverse leakage

</div>
(36)<div class='page_container' data-page=36>

Summary

of

transport

analysis

• ENDF : Evaluated Nuclear Data Library
• resonance parameters

• pointwise cross sections
• Energy condensation

• resonance integral

• scattering matrix

• CPM – multigroup condensation to multigroup (69~100)
library

• MOC – assembly wise calculation to obtain few group (2~10)
constant for core wide calculation

• Diffusion method – isotropic scattering/ angle independent
flux

• Core wide analysis

 Monte Carlo method – usually for reference cases

 SN method – when non-isotropy is important, such as
shielding analysis

ENDF

69 group
library

few group
constant

power distribution,
reactivity coefficient,

etc.

Diffusion;
Nodal method

Transport
calculation;

MOC

Energy, Temperature
effect:

CPM

Monte Carlo

reference

point library

</div>

Lý thuyết Lò phản ứng

2. Neutron interaction and

transport

Neutron

interaction

<sub>U</sub>

<i>n</i>



<i>X</i>



<i>X</i>

 





 

number of ptl. scattered into solid angle

per unit time

incident intensity

<i>d</i>

<i>d</i>



  



2 sin

<i>d</i>

 



 

<i>d</i>

<i>s</i>

<i>ds</i>

<i>d</i>







<i>dI</i>



<i>N I</i>

<i>N Idx</i>



 



 

<i>I x</i>



<i>I e</i>

<i>N</i>



 

 

<i>I x</i>



<i>I e</i>

















Macroscopic

cross

section

Monte Carlo simulation method

Neutron

transport







 



Neutron

transport

equation

‐

derivation





