Lesson 2 Objectives

1
Lesson 2 Objectives

• Fission chain reactor
• 4 factor formula
• Time dependence concepts
• Diffusion concepts

2
Chapter 2 Neutron Chain Fission Reactors

• Mainly 4 factor formula. Keep these straight

3
Integral form of 4-factor formula
4
Time dependence

• Introduce N0 neutrons at time0
• Solve it for N(t) at 100 lifetimes with S0,
  k1.005

5
Steady-state with constant source

• Source multiplication
• Basis for 1/M method of reactor startup

6
Effect of delayed neutrons
7
Effect of delayed neutrons (2)

• Note similarity to source equation except
• (k-1) replaced with (k-b-1)
• k vs. 1b is measure of prompt criticality
• Precursor concentration rises (slowly) with time

8
Prompt-jump (-drop) approximation

• The prompt jump and drop assume that the
  neutron flux will instantly respond to k(t) at
  the current source level of C(t). Set the
  neutron derivative to 0
• Substitute into C(t) equation
• This exponential sets the pace for power rise
  (or fall)

9
Diffusion Theory

• Diffusion theory Simplified treatment of
  direction components of neutron flux
• Same spatial (finite difference) and energy
  (multigroup) treatments as transport theory
• One speed Multiple groups coupled through
  scattering and fission

10
Diffusion Theory (2)

• Of more importance to us are assumptions
• Uniform medium
• Isotropic scattering
• Slowly varying flux
• Absorption small compared to scattering
• No sources nearby
• Results Ficks law
• General reasonableness of Ficks Law?

11
Derivation of Diffusion Theory from Transport
Theory

12
Derivation of Diffusion Theory from Transport
Theory (2)

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Derivation of Diffusion Theory from Transport
Theory (3)

• Resulting equation for L1
• Form the kth flux moment equations

14
Derivation of Diffusion Theory from Transport
Theory (4)

• If we set , we get the coupled
  equations
• or

15
Derivation of Diffusion Theory from Transport
Theory (5)

• If we define the absorption cross section as
• And the diffusion coefficient as
• This becomes

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Interface and Bound. Cond.s

• Partial currents
• Interface conditions
• Boundary conditions
• Reflected

17
Vacuum Boundary Conditions

• Rigorous
• Can use directly (mixed)
• Can convert to an approximate extrapolated
  0-flux
• Can ignore the extrapolation

18
One-Speed Diffusion Equation

• The diffusion equation in 3D is
• Questions
• What is the meaning of the vector operators?
• Where is scattering? What order is it?
• Why was fission separated out?
• What are the limitations on the source?
• Inside a homogeneous region

19
Applicability of Diff. Theory

• Return to the assumptions
• Uniform medium
• Isotropic scattering
• Slowly varying flux
• Absorption small compared to scattering
• No sources nearby
• How can we apply it to a reactor?

20
Diffusion migration lengths

• LDiffusion length
• L21/6 mean squared distance traveled by thermal
  neutron to captureD/Sa
• tage to thermal1/6 mean squared distance
  traveled by fast neutron to thermal
• Mmigration length
• 1/6 mean distance squared
  traveled by a
• neutron from birth to
  absorption
• Table 3.1 and the kernels of Equation 3.26

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Homework 2

• Assume you have a critical reactor with
• At t0, the k instantaneously jumps to 10 cents
  supercritical (k10.1b)
• What will be the time behaviour of N(t), assuming
  the prompt-jump approximation?
• Check the PJ approximation What is the
  approximate time that it took for the neutron
  population to make 99 of the jump?

22
Homework 2 (contd)

• Work problems 2.1 and 3.4 from the text
• Give a complete derivation of diffusion theory
  from transport theory (Slides 11-15) by including
  the integrations I skipped
• Add a column to Table 3.1 for the mean distance
  traveled by a neutron
• From birth to thermalization
• From thermalization to absorption
• Total (from birth to absorption)

23
Homework 2 (contd)

• Using the point kernel for the flux (Eq. 3.26)
  and remembering that
• Absorption profile is flux time absorption
  cross-section
• dV4pr2dr for a sphere about a point
• Show that L squared is 1/6 of average distance
  travelled from birth (at origin) to absorption
  squared, i.e.,
• What is the average distance travelled from birth
  to absorption (in terms of L)?