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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

transportJonghwa Chang jhchang@kaeri.re.kr

(n,3n)

elastic scatteringcapture (n,)

fission (n,f)

(n,n’)

(n,2n)fission spectrumcross section plot: http://atom.kaeri.re.kr/

scattered neutron

(n,n) – elastic scattering

scattering

Compound reaction

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

Monte Carlo simulation method

- Tracking individual neutron flight and collision history

- statistical average behaviour

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

GEANT : high energy, inaccurate cross sectionEGS-4, Penelope : electron-gamma shower

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

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 )

Neutron flux : r, , ,E  t vnr, , ,E  t

- 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

Recall definition of flux  vn

Time dependent neutron transport equation

Chain reaction problem 1 f ext

Difficulties

- Energy dependent cross section is widely varying

- resonance cross section is dependent on temperature

Neutron slowing down – mostly by elastic scattering

 

2 2

2 '2

Average logarithmic energy decrement

Average number of collisions

practical highest energy : E0= 20 MeV

Good moderator material

enriched uranium is needed natural uranium can be used

thick reflector is needed ~ 1meter.

(reactor is bulky)

Characteristics of typical moderator

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

E s s

Infinite homogeneous medium

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

   

/

s

q E

E m

speed distribution

Thermal energy

Thermal equilibrium of atoms follows Maxwell Boltzman distribution

Neutron scattering with medium in thermal motion

* important for moderator, esp cold neutron source

Ref) INDC-NDS-0475 (IAEA 2005)

free gas model

Bragg’s cutoff

Bragg’s peak

   

 

0 ,

0

, ,

3/ 2

2 2 E k T B

n B

• resonance appears at very high energy : ~MeV

• energy width is wide

• amplitude is not strong

Neutron resonance

upper bound

Breit-Wigner single level resonance formular

de Broglie wave length of neutron

For low energy E~0

2 0 0

1 4 1

l l

P E E

0 2

1 4 1

l E E

1 1

E E

1/ 2 0

contribution from all s-wave resonance tails

Average absorption in resonance energy range

• (broad) flux is 1/E outside narrow region of resonance

(constant in lethargy unit)

   

 

a resonance res

a res

a res

res

RI u

1 1

E E

Target nucleus are moving due to temperature

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

4 2

B

MV k T B

1 1

r

r

E E

When T is low,  is large

1 ,

2

1 exp 4 ,

1 2

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)

Scattering resonance at fuel(resonance) nuclide is small

• NR (Narrow Resonance) approximation

Large background = infinite dilution

 

R a

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

Solve NxN system of equation for angular direction

Spatial discretization : FDM, etc

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

streaming problem

Pij can be derived analytically (for simple geometry)

Widely used for multigroup condensation

' 3 2

analytic solution exists

DeCART : developed by SNU, Korea

OpenMOC

Ray-tracing method

s





0 : absorption cross section

1 : transport cross section

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 leakageQUABOX, CUBOX (KWU) : polynomial expansions

MASTER (KAERI) : Nodal expansion method

Powerful when scattering, flux can be considered as isotropic : Large LWR problem

• 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

• Diffusion Equation Formulation

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

current

get nodal balance equation



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

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

Many variants proposed

• 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;