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### Bessel Functions— Very elementary considerations —

Nuclear Reactor Theory

Intermezzo:

One-Speed Diffusion Theory

of a Nuclear Reactor

See Duderstadt & Hamilton: § 5.III AD

§ 5. IV

[to a large extent to be studied independently]

William D’haeseleer

BNEN – NRT 2011-2012

Nuclear Reactor Theory1. One speed (mono-energetic) reactor equation

Consider mono-energetic neutrons only

→ time-dependent neutron diffusion equation

Consider s as the neutron source due to fission

William D’haeseleer

BNEN – NRT 2011-2012

Nuclear Reactor TheorySimple Geometry:Slab Reactor

Consider steady state: critical system

slab geometry; thickness a

a

William D’haeseleer

BNEN – NRT 2011-2012

Nuclear Reactor TheorySimple Geometry:Slab Reactor

Solution:

Because of symmetry: C = 0

a

Second b.c.:

Non-trivial solution: n: odd integer

William D’haeseleer

BNEN – NRT 2011-2012

Nuclear Reactor TheorySimple Geometry:Slab Reactor

For a critical reactor, only 1-st “harmonic” remains:

a

Since:

“buckling”

For “a” very large, B1 very small, almost no buckle

William D’haeseleer

BNEN – NRT 2011-2012

Nuclear Reactor TheorySimple Geometry:Slab Reactor

Customary to designate

At any time in this case:

“Geometric Bucling”

William D’haeseleer

BNEN – NRT 2011-2012

Nuclear Reactor TheorySimple Geometry:Slab Reactor

Boundary conditions specify eigenvalues; constant A remains undetermined.

Must be found from overall power considerations

a

William D’haeseleer

BNEN – NRT 2011-2012

Nuclear Reactor TheoryOther simple reactor shapes

Sphere

and Φ finite everywhere

Solution:

William D’haeseleer

BNEN – NRT 2011-2012

Nuclear Reactor TheoryOther simple reactor shapes

Infinite cylinder

and Φ finite everywhere

Zero’s of J0; J0(xn)=0

Solution:

William D’haeseleer

BNEN – NRT 2011-2012

Nuclear Reactor TheoryOther simple reactor shapes

- Bessel functions J0, Y0
- Intermezzo on Bessel functions / following slides

–see also text “Bessel functions and their relatives” (GVdB & WDH)

William D’haeseleer

BNEN – NRT 2011-2012

William D’haeseleer

2007-2008

William D’haeseleer

BNEN – NRT 2011-2012

Intermezzo Bessel Functions

Solution of in rectangular coordinates

Take r as 1-D variable:

William D’haeseleer

BNEN – NRT 2011-2012

Intermezzo Bessel Functions

- Series expansion of Cosine and Sine functions

William D’haeseleer

BNEN – NRT 2011-2012

Intermezzo Bessel Functions

Solution of in cylindrical coordinates

Take r as 1-D variable:

(Note:

error in pdf document)

J0 and Y0 are Bessel functions of 0-th order

William D’haeseleer

BNEN – NRT 2011-2012

Intermezzo Bessel Functions

- Series expansions of zero-th order Bessel Functions

William D’haeseleer

BNEN – NRT 2011-2012

Intermezzo Bessel Functions

Asymptotic approximations (x sufficiently large)

For x large, each behaves as “damped” cos and sin

by sqrt(x)

William D’haeseleer

BNEN – NRT 2011-2012

Intermezzo Bessel Functions

Solution of in rectangular coordinates

William D’haeseleer

BNEN – NRT 2011-2012

Intermezzo Bessel Functions

Solution in cylindrical coordinates

Take r as 1-D variable:

(Note:

error in pdf document)

I0 and K0 are Modified Bessel functions of 0-th order

William D’haeseleer

BNEN – NRT 2011-2012

Intermezzo Bessel Functions

Modified Bessel Functions of 0-th order

William D’haeseleer

BNEN – NRT 2011-2012

Intermezzo Bessel Functions

Asymptotic approximations (x sufficiently large)

For x large, each behaves as “reduced” exponentials or hyperbolic functions

by sqrt (x)

William D’haeseleer

BNEN – NRT 2011-2012

Nuclear Reactor TheoryOther simple reactor shapes

Sphere / Infinite cylinder / Parallelepiped / Finite cylinder

William D’haeseleer

BNEN – NRT 2011-2012

Nuclear Reactor TheoryOther simple reactor shapes

William D’haeseleer

BNEN – NRT 2011-2012

Ref: Lamarsh NRT

Nuclear Reactor TheoryOther simple reactor shapes

William D’haeseleer

BNEN – NRT 2011-2012

Ref: Lamarsh NRT

Nuclear Reactor Theory1. One speed (mono-energetic) reactor equation

Evolution of flux dependent upon

- geometry (diffusion; leakage)
- material composition (absorption; fission)

→ For arbitrary geometry & composition,

difficult to find Bg ; also in general:

Steady state only arises when reactor is critical;

i.e., when k = 1

William D’haeseleer

BNEN – NRT 2011-2012

Nuclear Reactor Theory1. One speed (mono-energetic) reactor equation

In general, k ≠ 1;

therefore apply a “trick” to find k = f (dimensions, material composition)

then set k = 1 relationship between dimensions & material composition for criticality

!

William D’haeseleer

BNEN – NRT 2011-2012

Nuclear Reactor Theory1. One speed (mono-energetic) reactor equation

≡ B² called“Material Buckling” Bm2

-- for k = 1 --

At this stage, k still unknown, since B² is not known

William D’haeseleer

BNEN – NRT 2011-2012

Nuclear Reactor Theory1. One speed (mono-energetic) reactor equation

Note (1):

- suppose one writes

→ one could introduce f = fuel utilization factor

William D’haeseleer

BNEN – NRT 2011-2012

Nuclear Reactor Theory1. One speed (mono-energetic) reactor equation

Note (2):Consider an infinite reactor

in such a reactor, Φ independent of position

(uniform throughout)

reduces to

William D’haeseleer

BNEN – NRT 2011-2012

Nuclear Reactor Theory1. One speed (mono-energetic) reactor equation

Note (3):Consider now general case (finitereactor)

- Take now as a definition:

Then, source term

→ time-(in)dependent diffusion equation can be written as:

Time independent

Time dependent

William D’haeseleer

BNEN – NRT 2011-2012

Nuclear Reactor Theory1. One speed (mono-energetic) reactor equation

Note (4):still finite reactor

- Time independent case:

Or for a critical reactor, k = 1

William D’haeseleer

BNEN – NRT 2011-2012

Nuclear Reactor TheoryMono-energetic critical equation

Reactor is critical if

Material Buckling

Geometric Buckling; B12=Bg2

for simple slab

“critical equation”

William D’haeseleer

BNEN – NRT 2011-2012

Nuclear Reactor TheoryMono-energetic critical equation

“critical equation”

determines requirement to have a critical reactor:

- material composition

- dimensions etc.

determines requirement to have a critical reactor:

- material composition

- dimensions etc.

Alternative expression of critical equation:

William D’haeseleer

BNEN – NRT 2011-2012

Nuclear Reactor TheoryMono-energetic critical equation

Physical meaning of critical equation

Consider a bare reactor of arbitrary geometry.

# of neutrons leaking out of the system

# of neutrons absorbed

William D’haeseleer

BNEN – NRT 2011-2012

Nuclear Reactor TheoryMono-energetic critical equation

Physical meaning of critical equation

Non-leakage probability*:

From:

William D’haeseleer

BNEN – NRT 2011-2012

*Note: symbols PL and PNL are used interchangeably and mean the same!

Nuclear Reactor TheoryMono-energetic critical equation

Physical meaning of critical equation

Interpretation of

Physical meaning of critical equation

Interpretation of

# of neutrons absorbed

Gives rise to a release of fission neutrons:

William D’haeseleer

BNEN – NRT 2011-2012

Nuclear Reactor TheoryMono-energetic critical equation

Physical meaning of critical equation

Physical meaning of critical equation

Of these neutrons, only a fraction PL does not leak out, and gives rise to absorption in the next generation:

Hence:

For a critical reactor:

William D’haeseleer

BNEN – NRT 2011-2012

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