Flux Shape in Various Reactor Geometries in One Energy Group

1
Flux Shape in Various Reactor Geometries in One
Energy Group

• B. Rouben
• McMaster University
• EP 6D03
• Nuclear Reactor Analysis
• 2009 Jan-Apr

2
Contents

• We derive the 1-group flux shape in the critical
  homogeneous infinite-slab reactor and
  infinite-cylinder reactor Exercises for other
  geometries.
• Reference Duderstadt Hamilton Section 5 III

3
Diffusion Equation

• We derived the time-independent neutron-balance
  equation in 1 energy group for a finite,
  homogeneous reactor
• We showed that we could introduce the concept of
  geometrical buckling B2 (the negative of the flux
  curvature), and rewrite the equation as
• where B2 has to satisfy the criticality condition

4
Solving the Flux-Shape Equation

• Eq. (2) is the equation to be solved for the flux
  shape.
• We will study solutions of this shape equation
  for various geometries, and will start with the
  case of an infinite cylindrical reactor.
• The thing to remember is that the solution must
  satisfy the diffusion boundary condition, i.e.
  flux 0 at the extrapolated outer surface of the
  reactor.
• While Eq. (2) can in general have a large
  multitude of solutions, we will see that the
  addition of the boundary condition makes Eq. (2)
  an eigenvalue problem, i.e., a situation where
  only distinct, separated solutions exist.
  

5
Solving the Flux-Shape Equation

• We will now apply the eigenvalue equation to the
  infinite-slab geometry and the infinite-cylinder
  geometry, and solve for the geometrical buckling
  and the flux shape.
• As concluded before, we will always find that the
  curvature of the 1-group flux in a homogeneous
  reactor is always negative.

6
Interactive Discussion/Exercise

• Given that the curvature of the 1-group flux in a
  homogeneous reactor is negative, where do you
  think the maximum flux would have to be in a
  regular-shaped reactor?
• Explain.

7
Maximum Flux

• In a 1-group homogeneous reactor of regular
  shape, the maximum flux must necessarily be at
  the centre of the reactor.
• We can explain that by reductio ad absurdum.
  Suppose the flux is a maximum not at the centre
  of the reactor if we draw a straight line from
  the maximum point to the centre of the reactor,
  then by symmetry, there would have to be another
  maximum on the same line on the other side of
  the centre. Therefore the centre of the reactor
  would be at a local minimum along that straight
  line, which would imply that the flux does not
  have negative curvature along that straight line!

8
Infinite-Slab Case

• Let us study the simple case of a slab reactor of
  width a in the x direction, and infinite in the y
  and z directions.
• Eq. (2) then reduces to its 1-dimensional
  version, in the x direction
• 
  (4)
• We can without loss of generality place the slab
  symmetrically about x 0, in the interval -a/2,
  a/2.
• 
  contd

9
Infinite-Slab Reactor Geometry
x -a/2
x a/2
x aex/2
x 0
10
Infinite Slab (contd)

• We also know that
• The flux must be symmetric about x 0, and
• The flux must be 0 at the extrapolated
  boundaries, which we can call ? aex/2
• Eq. (4) has the well-known solutions sin(Bx) and
  cos(Bx).
• Therefore the most general solution to Eq. (4)
  may be written as
• 
  (5)
• 
  
  contd

11
Infinite Slab (contd)

• However, symmetry about x 0 rules out the
  sin(Bx) component.
• Thus the reactor flux must be
• 
  (6)
• We can determine B from the boundary condition at
  aex
• (7)
• Now remember that the cos function has zeroes
  only at odd multiples of ?/2. Therefore B must
  satisfy
• (8)
• 
  
  contd

12
Infinite Slab (contd)

• It looks as if there is an infinite number of
  values of B.
• While that is true mathematically, the only
  physically possible value for the flux in the
  critical reactor is the one with the lowest value
  of B, i.e. for n 1
• 
  (9)
• We can conclude that Eq. (9) is the only
  physical solution from the fact that the
  solutions with n 3, 5, 7, all feature regions
  of negative ? in the reactor, and that is not
  physical.
• Also to be noted from Eq. (9) is that the
  buckling increases as the dimensions of the
  reactor (here aex) decrease as had also been
  concluded earlier the curvature needs to be
  greater to force the flux to 0 at a closer
  boundary!
  
  
  
• 
  
  contd

13
Infinite-Slab Case (contd)

• The 1-group flux in the infinite-slab reactor can
  then be written
• 
  
• The absolute value of the flux, which is related
  to the constant A1, is undetermined at this
  point.
• This is because Eq. (3) is homogeneous
  therefore any multiple of a solution is itself a
  solution. The physical significance of this is
  that the reactor can function at any power level.
• Therefore, to determine A1, we must tie the flux
  down to some quantitative given data e.g., the
  desired total reactor power.
  contd

14
Infinite-Slab Case (contd)

• Because the slab is infinite, so is the total
  power. But we can use the power generated per
  unit area of the slab as normalization. Let this
  be P W/cm2.
• If we call Ef the recoverable energy per fission
  in joules (and we know that this is 200 MeV
  3.210-11 J), then we can write
• 
  
• And doing the integration will allow us to find
  A1
• 
  
• i.e.,
  
• So that finally we can write the absolute flux as
  
• 
  

15
Additional Notes

• In solving this problem, we found that the
  reactor equation had very specific solutions,
  with only specific, distinct values possible for
  B.
• Actually, there is a general point to be made
  here Equations such as Eq. (3), holding over a
  certain space and with a boundary condition,
  i.e., the diffusion equation for the reactor,
  fall in the category of eigenvalue problems,
  which have distinct solutions eigenfunctions -
  here the flux distribution ? -, with
  corresponding distinct eigenvalues (here the
  buckling B2).
• Although in this problem we found only 1
  physically possible eigenfunction for the
  steady-state reactor (the fundamental
  solution), the other eigenfunctions are perfectly
  good mathematical solutions, which do have
  meaning.
• While these higher eigenfunctions (which have
  larger values of B) cannot singly represent the
  true flux in the reactor, they can exist as
  incremental time-dependent perturbations to the
  fundamental flux, perturbations which will die
  away in time as the flux settles into its
  fundamental solution.

16
Case of Infinite-Cylinder Reactor

• Infinite height Radius
  R

r
17
Infinite-Cylinder Reactor

• For a homogeneous bare infinite cylinder, the
  flux is a function of the radial dimension r
  only. All axial and azimuthal positions are
  equivalent, by symmetry.
• We write the eigenvalue equation
  in cylindrical co-ordinates, but in the
  variable r only, in which the divergence is
• 
  
• The 1-group diffusion equation then becomes
• 
  
• By evaluating the derivative explicitly, we can
  rewrite Eq. (16) as
• 
  
• 
  contd

18
Infinite-Cylinder Reactor (contd)

• We may be completely stumped by Eq. (17), but
  luckily our mathematician friend recognized it as
  a special case of an equation well known to
  mathematicians, Bessels equation (m is a
  constant)
• 
  
  
• Eq. (17) corresponds to m 0, for which this
  equation has 2 solutions, the ordinary Bessel
  functions of the 1st and 2nd kind, J0(Br) and
  Y0(Br) respectively.
• These functions are well known to mathematicians
  (see sketch on next slide)!
• It sure helps having mathematicians as friends,
  isnt it, even if Nobel didnt like them!
  contd

19
Infinite-Cylinder Reactor (contd)

• I sketch the functions J0(x) and Y0(x) below

Although, mathematically speaking, the general
solution of Eq. (17) is a combination of J0(x)
and Y0(x), Y0(x) tends to -? as x tends to 0 and
is therefore not physically acceptable for a
flux. contd
20
Infinite-Cylinder Reactor (contd)

• The only acceptable solution for the flux in a
  bare, homogeneous infinite cylinder is then
  
• 
  
• The flux must go to 0 at the extrapolated radial
  boundary .
• Therefore we must have
  
• The figure in the previous slide shows that J0(x)
  has several zeroes, labelled the 1st is at x1
  2.405, the 2nd at x2 ? 5.6
• But because, physically, the flux cannot have
  regions of negative values, B for the infinite
  cylinder can be given only by
• Therefore the buckling for the infinite cylinder
  is
• 
  
  

21
Infinite-Cylinder Reactor (contd)

• The 1-group flux shape in the infinite
  homogeneous cylindrical reactor is then
• 
  
• As before, the absolute magnitude of the flux
  (i.e., the constant A) can be determined only
  from some quantitative information about the
  flux, for example the power per unit axial
  dimension of the cylinder.
• If we denote that power density P, and the energy
  released in fission, we can write
• 
  

22
Infinite-Cylinder Reactor (contd)

• The integral on the Bessel function may look
  forbidding, but it can be evaluated from known
  relationships between various Bessel functions.
• Ill just give the final result here without
  derivation.
• If we ignore the extrapolation distance,
• which gives for the 1-group flux the final
  equation
• 
  

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Exercise/Assignment Apply to Other Shapes

• Exercise Apply the eigenvalue Eq. (3) to the
  following geometries to find the geometrical
  buckling and the flux shape
• Parallelepiped
• Finite cylinder
• Sphere
• Note in the cases of the parallelepiped and of
  the finite cylinder, invoke separability, i.e.,
  write the solution as a product of functions in
  the appropriate dimensions, each with its own
  directional bucklings, which add to the total
  buckling.

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Parallelepiped Reactor
y
b/2 -b/2
z
c/2
-c/2
x
-a/2 a/2
25
Parallelepiped Reactor

• Directional bucklings
• Total buckling
• Flux shape
• If total power P, and neglecting extrapolation
  distance

26
Finite-Cylinder Reactor

• Finite height
  Radius R

H/2 -H/2
r
27
Finite-Cylinder Reactor

• Directional and total bucklings
• Flux shape

28
Spherical Reactor

• Sphere of radius R
• Buckling
• where, for the same reasons as in the other
  geometries, B must take the lowest value allowed,
  to guarantee that there will not be regions of
  negative flux in the reactor.
• Flux shape
• 
  
• 
  

29
Flux Amplitude for a Spherical Reactor

• We can integrate Eq. (40) to evaluate the flux
  amplitude A for a given total reactor power P.
  If ER is the energy released per fission, and
  neglecting the extrapolation distance

2009 January
29
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Summary

• We can obtain the solution for the 1-group flux
  shape in bare homogeneous reactors of various
  geometries.
• In each case we determine directional bucklings
  (if applicable) and the total buckling, in terms
  of the dimensions of the reactor.
• The buckling(s) must take the lowest mathematical
  values allowed, to ensure that the flux solution
  is physical everywhere in the reactor.

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• END