HW

1
HW7

• Find the group absorption cross section for
  U238O2 for a 3-10 eV group using
• (a) Narrow resonance approximation
• (b) Wide resonance approximation
• For each of these, use
• Cross sections from 92238.dat and 8016.dat on
  public area
• Wigner rational approximation for PF0
• Dancoff factor of 0.2345
• d1 cm and UO2 density of 10 g/cm3.
• Previous slide definition of Gf

2
Lesson 8 Objectives

• Effects of irradiation and decay
• Reading decay chain charts
• Translating decay chain charts into differential
  equations
• Xenon and Samarium
• Burnable poisons

3
Effects of irradiation and decay

• The reactor composition changes with time due to
  irradiation effects and radioactive decay
• These effects must be taken into account in the
  design of the reactorin fact may be the primary
  design consideration of the reactor itself (e.g.,
  SRP production reactors, HFIR reactor)
• In this lesson we will write down the equations,
  discuss their solution, and examine the important
  decay chains
• In addition, we will look at the important design
  considerations brought on by burnup

4
Equations and solutions

• Reactor isotopic changes occur for two reasons
  irradiation effects (primarily due to fission and
  transmutation) and radioactive decay
• Figure 6.1 shows the transmutation-decay chains
  for U-238 and Th-232 (alpha decay ignored)
• Notice the overlap (i.e., the two charts fit
  together)
• Fertility from Th-232 and U-238

5
Eqns and solutions (2)

6
Eqns and solutions (3)

7
Eqns and solutions (4)

• In general, an equation can be written for each
  isotope in the reactor, with some combination of
  these terms
• Eqn. 6.3
• Note that this is given in the text for fission
  products, but is generally applicable to all
  isotopes
• Note also the flux terms as combining with
  decay constants (helpful to convert flux to
  /bn/sec and decay constants to 1/sec)

8
Eqns and solutions (5)

• The equations on pp. 199-201 of the text
  translate these relations into the
  transmutation-decay chains of interest to us,
  with some shortcuts
• Long-lived decay is ignored (e.g., alpha)
• Some short-lived decay is considered
  instantaneous
• The non-iterative linking of these equations
  allows for simple ODE chain solutions (See
  Example 6.1 on p. 203)

9
Solution of equations

10
Solution of equations

• For no source
• For constant source
• For exponential source

11
Eqns and solutions (6)

• More formally, the linked equations can be
  translated into a simple matrix ODE
• with solution
• Note that the burnup time steps are imposed
  because A and F depend on flux. The flux at ti
  is used for times up to ti1, when it is updated
  again
• The solution depends on this freezing of flux
  dependence and its accuracy will depend on the
  size of the time steps

12
Consequences of burnup

• Drop in k-effective as fuel burns out and fission
  products are built up
• Somewhat compensated for by Pu-239 build-in
• Unit of MWD/tonne of HM (consumption of 1 g
  U-235 1.07 MWD, therefore 100 enriched would
  burn to give 1 million MWD/tonne)
• Generally run BWRs and PWRs with fuel loadings
  that include different batches of fuel with
  differing burnups (at each shutdown replacing
  most burned-up with fresh fuel and reshuffling)

13

14

15
Consequences of burnup (2)

• Samarium and Xenon build-in and decay
• Both are poisons (Xe about 50x more absorptive
  per atom) and both reach an equilibrium at
  constant reactor flux level
• Reactivity held in Xenoneffect on k-effective
  of the steady-state Xe absorption
• After shutdown from steady state
• Sm goes to a higher equilibrium
• Xe rises to a peak and then decays away
• Reactivity effect important (especially Xenon) to
  restart times for reactors
• Reactivity effect also important for power level
  changes

16

17

18

19

20

21
Consequences of burnup (3)

• Burnable poisons Put in to flatten reactivity
  profile and extend reactor life

22
HW8

• In a Pu239 thermal system with f1014 n/cm2/s,
  what is the rate of production of Pu240 and Pu241
  in grams/cm3/day? (Assume Pu239 density is 0.01
  g/cm3.)
• Derive Equations 6.10, 6.11, 6.12, 6.16, 6.17,
  and 6.19 (plus the f threshhold in the note
  before 6.19).
• Estimate the U235 density (g/cm3) and flux
  (/cm2/sec) implied in Figures 6.5 and 6.6.