4/14/2015
Reactor Theory 1. One Group Reactor Equation a) Assumptions {bare reactor} (1) Rx is a homogeneous fuel & moderator (2) Rx consists of one ...
Reactor Theory 1. One Group Reactor Equation a) Assumptions {bare reactor} (1) Rx is a homogeneous fuel & moderator (2) Rx consists of one region and has neither a blanket or reflector. b) Diffusion Equation
Nuclear Reactor Theory Chapter 6
∅ − Σ ∅ + =
1
Eq. 1
D – Diffusion coefficient Σ - x section for absorption v- n yield S- source density (n/cm3 –s) = νΣ ϕ
Reactor Theory
2
Reactor Theory So we have 1 ∅ − Σ ∅ + Σ ∅ = 0
If the fission source neutrons do not balance the leakage and absorption terms, then the RHS of Eq. 1 is non-zero.
Let
= { Σ - Σ }
then ∅ = − ∅ substituting − ∅ for ∅
In this case we could balance the equation by multiplying the source term by a constant say “1/k ”.
We get − ∅ − Σ ∅ + Σ ∅ = 0 Eq.2
If the source is too small then k 1.
∅
Solving for k : = ∅
! ∅
=
!
3
Reactor Theory k = the multiplication factor
4
Reactor Theory Notice that = νΣ ϕ = ηΣ# ϕ = η !$ Σ ϕ !
∅
= ∅
! ∅
=
#n/fiss
!
Numerator = neutrons born via fission Denominator = neutrons lost to leakage and absorption.
f=
!$ !
Fuel f n per abs
= fuel utilization: the fraction of all
neutrons absorbed in the reactor that are absorbed in the fuel.
= η%Σ ϕ 5
Eq. 3. 6
1
4/14/2015
Reactor Theory C. Multiplication factor in the infinite reactor. Assumptions: 1. All neutrons are absorbed. No neutrons leak. 2. The neutron flux φ is constant everywhere. Is independent of position. Concept: 1. Since all neutrons born are eventually absorbed, then Σ ϕ is the total number of neutrons. 2. Of these %Σ ϕ are absorb in the fuel, and release η%Σ ϕ in the next generation
Reactor Theory Multiplication factor in the infinite reactor.
Reactor Theory D. Buckling B2 in the Critical bare reactor.
Reactor Theory D. Buckling B2 in the Critical bare reactor.
Concept: 3. Dividing the number of neutrons in one generation by those in the next gives effective yield/absorp & =
' ! ( ! (
= η%
Fuel utilization 4. & is k for and infinite reactor. 5. Since η and % are constants that depend on the material properties of the reactor, & is the same for a bare reactor as for an infinite reactor of the same composition. 6. Thus & refers to a Rx in which no neutrons leak.
7
1. Since & =
' ! ( ! (
= η%
8
4. If Rx is critical, k = 1, and
=
0,
− ∅ + (& −1)Σ ∅ =0 5. Let / = 0 ! = diffusion area, then (& − 1)∅ − ∅ + = 0 / 2 6. Solving for B (& − 1) = /
2. And the source term = η%Σ ϕ , then = & Σ ϕ 3. Using this in the one-group Rx eqn. {use Eqn. 2 & 3} 1 *+ − ∅ − Σ ∅ + = *, 1 *+ − ∅ − Σ ∅ + (& Σ ∅) = *, 4. If Rx is critical, k = 1, and
=
0,
− ∅ + (& −1)Σ ∅ =0 9
10
Reactor Theory
Reactor Theory
Ex 6.1 … Find f and k∞ for a mixture of U-235 and sodium (Na) in which the U is present to 1 w/0. Solution:
