Lý thuyết Lò phản ứng

QUABOX, CUBOX (KWU) : polynomial expansions MASTER (KAERI) : Nodal expansion method. Powerful when scattering, flux can be considered as isotropic : Large LWR problem.[r]

2 Neutron interaction and transport

Neutron interaction

(n,3n) elastic scattering

capture (n,)

fission (n,f)

(n,n’)

235U

1 *

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

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 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

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

1

x

xe dx     





Number density

A

N N

A 



  (gr/cm )3  1

barn 0.6022 cm

(u) A 

     

Macroscopic cross section

fission

moderation absorption

moderation leak

Speed of neutron :

v 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 /E m

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  

1

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

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 s , ' , ' , ', ', ' '

V  E E  E t dE d dr dEd



       

   r r

3 net outflow of neutrons

   

ˆ

ˆ , , , , , ,

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

 r  r

Gauss’s theorem 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)

- 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

0

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 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 

              , , , , , , , , , , 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

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

Slowing down

Neutron slowing down – mostly by elastic scattering

 

2

2 '2

2

'

c

A A

E v

E v A



 

 



Energy after elastic scattering Maximum energy transfer

Average logarithmic energy decrement

Average number of collisions

 2

1

1 ln

2

A A

A A

    



2 /

A  



 '  lnE E'/  N E E



 

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)

Assumptions :

• above thermal energy (> 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



    /  '   

' '

'

E s s

E

E

E E E dE

E



 





  





Slowing down density

    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





Epithermal spectrum

el

 is nearly constant (except near resonance)

0

u

EE e

   

0

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 



 

 

 3/ 1/ /

2 E k TB

B

N E dE

E e dE

N  k T





  2 

3/

/ 2

v v

4 v v

2

B

mv k T B

N d m

e d

N k T 

        v E m

speed distribution

Thermal energy

Thermal equilibrium of atoms follows Maxwell Boltzman distribution

Neutron scattering with medium in thermal motion

    / ' ', , , , ' b d E

E E T e S T

d dE kT E

           b

 bound atom scattering cross secion

  2 , ,

S T e

        

Free gas model

'

E E

kT

  

0

' '

E E EE

A kT 

   

energy transfer momentum transfer

A0 : Mass number of atom

E

'

E

0

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)

Scattering in graphite

Ref) INDC-NDS-0475 (IAEA 2005)

free gas model

Bragg’s cutoff

 

  1/2 / 3/

2

B

E k T

B

N E dE N E e dE

k T  





neutron flux  

 03/ /

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

      , , ,

a M n

a th

M n

E E T dE E T dE

             / 3/

2 E k TB n

B N

E dE Ee dE

m k T     

   2 2 /

0

B

E k T

B B

Ee dE k T k T

      293 B a a k E  

  1/ , 0 293 293

a th a B B

a

k k T

T             / / 0 3/

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

0

a

 cross section at 2200 m/s

Maxwell average cross section at temperature T

,

a th



Recall gamma function

Heavy nuclides

• resonance energy region : ~ 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

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

v

na

E

E E

   

2

4 potential scattering term

s a

   

n E

 

2

2

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

238U

 

0

n

E g  



      

 

 at resonance peak

    0

2.608 10

4 b eV A E A              2 n E g E E          

near resonance peak        2 0 l l P E E P E E E                           

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

(exact for s-wave)

  1/ 2 

0 1 E E E y           

for s-wave resonance

 0

2

y EE



Resonance tail (E ~ 0, s-wave)

1/

0 2

0

1

1 /

E

E E E

 

     

    

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

  1/ 2 

0 2 1 E E E y              res E RI dE E    

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

 

/

2 2 / 2

1 1

1 tan cos

1 y dy d



    

 

     

 

 0

2

y EE



2 dEdy

0

0

1

2res1

RI dy E y          0 RI E      

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

r z x y

m m

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

 

2

r z

m

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

Doppler broadening

  2 

3/ / 2 B

MV k T B

M

P d V e d

k T 





 

  

 

V V V

assume Maxwell distribution of target nucleus

   

,T ,

E E x E       

 Doppler line shape function

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

 ,x

When T is low, is large

 

1 , x x    

When T is high, is small

  2

, exp

2

x  x

      

 

Line shape function for scattering cross section

     

2

2

exp /

,

1

y x y

x dy y            

Doppler line shape function

U-238 0K 300,000K     2 exp , x y x dy y                 

Line shape function for absorption cross section

 0

2

r

y E E



 0

2

x EE



4k TEB 4k TEB r

A A

  

  

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

    /    

' ' '

E

t E E E s E E E dE



   

   

    /   '  /   ' 

' '

' '

F M

F M

E E

s s

F F M

a s s

E E

F M

E E

E dE dE

E E

     

   

 

   

 

 

   

0

1

1 /

u a

E

E E

 

 

 

What is flux shape at resonance ?

pot



a



E

resonance

(U-238)

Scattering resonance at fuel(resonance) nuclide is small       /      ' ' ' 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 ' ' '

R R

R

E pot E pot

s u

u

E E

E

E dE dE

E E E E

     

 

   

 

 

   Rpot b

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 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

R t b E E E           R a WR b R

t b res E RI dE E       

Discrete Ordinate method (SN method)

Neutron transport equation (steady state)

Angular quadrature

   

0

2

' '

4

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 

 

4

1

N

n n n

fd w f





  

 n : ordinate

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

transport collision scattering source

n

 

1

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

Spherical harmonics (PN) method

Angular flux is approximated by a finite spherical harmonics expansion

      , 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

0

1

2

n n

n n n

d d

n n

q

n dx n dx

      

   

  isotropic source term

1

1

1

2

n n n n

d n n

dx n n

                 

Elliminate odd order terms

1

1 2 0

1 1

2 n n n n n 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 2 0

1 1

2 n n n n n n n n n

n n n n n n

q

n  n  n  n  n  n    

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

SP7 have equations

   

1

sn Pn s d

    

 

0 t s0 a

   

for

n sn n

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

1

ˆ

4  r d t r  r q r    

'

3

1

ˆ ' '

4 '

t

e

q d







 





 r r

r r r

r r

CPM

  ,      

1

,

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

'

1

'

4 '

t

ij

e

P d





 





 r r r

r r

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

Diffusion equation

SP1 form

1

1 0 0

1

3    q



    

Fick’s law

J   D 

J  q

   

Diffusion equation

1

1

D





0 : absorption cross section

1 : transport cross section

 

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

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

Nodal Method for Neutron Diffusion Equation

• Diffusion Equation Formulation

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

∙ Σ

Σ Σ →

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

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 x x x d x

D x q L x

dx        

xs Jxs Jxs

    

Partial current

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

  22  , x

y

D

L x x y dy

h y

  



Nodal Method for Neutron Diffusion Equation

   

x xn n

n

x a p x

  x  xn n 

n

L x b p x

 

0

1 for

1

0 for

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

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