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CH.II: NEUTRON TRANSPORT. INTRODUCTORY CONCEPTS ASSUMPTIONS NEUTRON DENSITY, FLUX, CURRENT REACTION RATE FLUENCE, POWER, BURNUP TRANSPORT EQUATION NEUTRON BALANCE BOLTZMANN EQUATION CONTINUITY AND BOUNDARY CONDITIONS INTEGRAL FORMS FORMAL SOLUTION USING NEUMANN SERIES.

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Ch ii neutron transport l.jpg
CH.II: NEUTRON TRANSPORT

INTRODUCTORY CONCEPTS

  • ASSUMPTIONS

  • NEUTRON DENSITY, FLUX, CURRENT

  • REACTION RATE

  • FLUENCE, POWER, BURNUP

    TRANSPORT EQUATION

  • NEUTRON BALANCE

  • BOLTZMANN EQUATION

  • CONTINUITY AND BOUNDARY CONDITIONS

  • INTEGRAL FORMS

  • FORMAL SOLUTION USING NEUMANN SERIES


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II.1 INTRODUCTORY CONCEPTS

ASSUMPTIONS

  • Interactions between n – matter: quantum problem

    But (n) << characteristic dimensions of the reactor E

  • Density of thermal n: ~ 109 n/cm3

    Atomic density of solids: ~ 1022 atoms/cm3

  • Interactions n – n negligible

  • Linear equation for the neutron balance

  • Statistical treatment of the n, but small fluctuations about the average value of their flux

     n: classical particles, not interacting with each other, whose average value of their probability density of presence is accounted for


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NEUTRON DENSITY, FLUX, CURRENT

Variables

Position: 3

Speed: 3 or kinetic energy + direction

(Time: 1 t)

Angular neutron density

: nb of n in about with a speed in [v,v+dv] and a direction in about

Neutron density

Angular neutron density whatever the direction

(same definition with variables (r,E,))

(dimensions of N in both cases?)


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[dim ?]

Angular flux

s.t.

Total neutron flux Integrated flux

(Angular) current density

 Nb of n flowing through a surface  / u.t. (net current)

Isotropic distribution

[dim ?]

Net current = 0, hence flux spatially cst?

Wrong!!

  • A reactor is anisotropic

  • But weak anisotropy (1st order)


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REACTION RATE

Nb of interactions / (volume.time): R

Beam of incident n on a (sufficiently) thin target (internal nuclei not hidden):

 R = N.(r,t).(r) = (r).(r,t)

Or: interaction frequency = v [s-1]

n density = n(r,t) [m-3]

 R = n(r,t).v(r) = (r).(r,t)

General case: cross sections dependent on E ( v)

Rem:  = f(relative vbetween target nucleus and n) while  = f(absolute vof the n)  implicit assumption(for the moment): heavy nuclei immobile

(see chap. VIII to release this assumption)

R =

Nb nuclei

cm3

Nb n

cm2.s

Cross sectional area

of a nucleus (cm2)

x

x


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Rem: differential cross sections

Scattering  speed after a collision?

  • Conditional probability that 1 n with speed undergoing 1 collision at leaves it with a speed in [v’, v’+dv’] and a direction in about

    with

  • Scattering kernel s.t.

    Isotropic case:

Why?

[dim ?]


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FLUENCE, POWER, BURNUP

Fluence

  • Characteristics of the irradiation rate ([n.cm-2] or [n.kbarn-1])

    Power

    Linked to the nb of fission reactions

    Burnup

    Thermal energy extracted from one ton of heavy nuclei in fresh fuel

     Fluence x <fission cross section> x energy per fission


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II.2 TRANSPORT EQUATION

dS

V

NEUTRON BALANCE

Variation of the nb of n (/unit speed)

in volume V, in dv about v, in about

Sources

Losses due to

all interactions

Losses through

the boundary

 Gausstheorem

(n produced in about ,

dv about v, about )

(n lost in about ,

dv about v, about )


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Rem: general form of a conservation equation

BOLTZMANN EQUATION (transport)

Without delayed n

Sources?

Steady-state form

Fraction in dv about v, d about 

Total nb/(vol. x time) of n due to all fission at r

Fission

Scattering

External

source


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Compact notation

with

(destruction-scattering operator)

and

(production operator)

Non-stationary form


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Fraction/(vol. x time) of n due

to all fissions at r in group i

With delayed n

Concentration Ci of the precursors of group i:

with i = (ln 2) / Ti

Def: production operator for the delayed n of group i, i = 1…6:

Total production operator:

Let

Radioactive decay

Production of delayed n of group i / (vol. x time)

prompt n

Fraction in dv about v, d about 


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System of equations for the transport problem with delayed n

Stationary regime

Reduction to 1 equation:

Production operator: equivalent to having J  Jo (prompt n) iff

 Formalism equivalent with or without delayed n


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CONTINUITY AND BOUNDARY CONDITIONS

Nuclear reactors: juxtaposition of uniform media (  indep. of the position)  How to combine solutions of

in the  media?

Let : discontinuity border (without superficial source)

Integration on a distance [-,] about in the direction

continuity on 

Boundary condition

(convexreactor surrounded

by an  vacuum):


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INTEGRAL FORMS

If s = distance covered in the direction of the n:

Lagrange’s variation of constants:

Let

(interpretation ?)

: optical thickness (or distance) [1]


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Yet

If both scattering and independent source are isotropic

  • After integration to obtain the total flux:

    Rem: S fct of  !!



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Transition kernel

 Transport process: proba distribution of the ingoing coordinates in the next collision, given the outgoing coordinates from the previous one

Collision kernel

 Impact: entry in 1 collision  exit

Compact notation :

(based on the negative exponential law)

  • = 1for an infinite reactor

  • Captures not considered

  • Fissions :   1


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Collision densities

Ingoing density: = expected nb of n entering/u.t. in a collision with coordinates in dP about P

Outgoing density: = expected nb of n leaving/u.t a collision with coordinates in dP about P

Evolution equations

Interpretation ?


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Rem:

  • equ. of (P) = (equ. of (P)) x t(P)

  • Possible interpretation of n transport as a shock-by-shock process

    FORMAL SOLUTION USING NEUMANN SERIES

    Let

     j(P): ingoing collision density in the jth collision

     : solution of the transport equation

  • Not realistic: infinite summation…

  • Basis for solution algorithms