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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

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

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?)

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)

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

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 ?]

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

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 )

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

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

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

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):

INTEGRAL FORMS

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

Lagrange’s variation of constants:

Let

(interpretation ?)

: optical thickness (or distance) [1]

Yet

If both scattering and independent source are isotropic

- After integration to obtain the total flux:

Rem: S fct of !!

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

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 ?

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

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