Kinetics With Delayed Neutrons

1
Kinetics With Delayed Neutrons

• B. Rouben
• McMaster University
• EP 4P03/6P03
• 2008 Jan-Apr

2
Point-Kinetics Equations

• In a previous presentation, we derived the
  point-kinetics equations, which govern the time
  evolution of the neutron density n and that of
  the delayed-neutron-precursor concentrations, Cg

3
Case without Delayed Neutrons

• ? is a very short time
• ? ? 0.9 ms in CANDU
• ? ? 0.03 ms in LWR
• It is easy to see that if there were no delayed
  neutrons at all, the point-kinetics equations
  would reduce to
• Thus, without delayed neutrons, the neutron
  density would grow (or drop) exponentially as

4
Case with Delayed Neutrons

• Delayed neutrons change this significantly.
• To solve the point-kinetics equations, we can try
  exponential solutions of the form

5
The Inhour Equation

• We can divide by n to get an equation for ?.
• Eq. (6) is usually recast into another form (the
  Inhour equation) by substituting

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Inhour Equation

• The Inhour equation is a complicated equation to
  solve in general, as the left-hand side is a
  discontinuous function which goes to ?? at
  several points.
• e.g., for G 6 from Duderstadt Hamilton

7
General Solution

• There are (G2) branches in the graph of the
  Inhour equation, and (G1) roots for ?
  (intersections with line ?).
• If ? lt 0 all roots will be negative, but
• If ? gt 0 one root will be positive (?1), and all
  other roots will be negative
• The general solution for n and Cg is then a sum
  of (G1) exponentials

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General Solution (cont.)

• By convention, we denote ?1 the algebraically
  largest root (i.e., the rightmost one on the
  graph)
• ?1 has the sign of ?.
• Since all other ? values are negative (and more
  negative than ?1 if ?1 lt0), the exponential in ?1
  will survive longer than all the others.
• Therefore, the eventual (asymptotic) form for n
  and Cg is exp(?1t), i.e., the power will
  eventually grow or drop with a stable (or
  asymptotic) period .

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General Solution (cont.)

• In summary, for the asymptotic time dependence
• For ? not too large and positive (i.e., except
  for positive reactivity insertions at prompt
  criticality or above)
• , i.e., things evolve much more
  slowly than without delayed neutrons

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Solution for 1 Delayed-Neutron-Precursor Group

• If we assume only 1 delayed-neutron-precursor
  group, the Inhour equation becomes a bit simpler

11
Solution for 1 Delayed-Neutron-Precursor Group
(cont.)

• Now there are 3 branches and 2 roots for ?
• When ? lt 0, both ? values are negative
• When ? gt 0, one ? value is positive, and the
  other is negative.
• When ? gt 0, one ? value is 0, the other is
  negative.
• Again, we label the algebraically larger one ?1.
• With 1 precursor group, the equation for ? can
  also be written as a quadratic equation
• which can be solved exactly

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Solution for 1 Delayed-Neutron-Precursor Group
(cont.)
13
Solution for 1 Delayed-Neutron-Precursor Group
(cont.)

• If we substitute the form
  into the point-kinetics equations, we get
  
• The general solution is

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Solution for 1 Delayed-Neutron-Precursor Group
(cont.)

• Using the values of ?1 and ?2 from Eqs. (14)
  (13)
• The 2nd term decays away very quickly (typically
  in 1 s), therefore the neutron density (or
  flux/power) experiences a prompt jump or drop by
  a factor ?/(?-?) this is good as long as ? is
  not too large with respect to ?

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Prompt Jump or Drop

• Illustration of prompt jump prompt drop is
  similar
• (Lamarsh Fig. 7.4)

16
Relationships at Steady State

• The point-kinetics equations apply even in steady
  state, with ?0.
• The relationship between the precursor
  concentrations and the neutron density can be
  obtained by setting the time derivatives to 0 in
  the point-kinetics equations. For G precursor
  groups at steady state (subscript ss)
• From Eq. (21) we get
• Note This relationship holds also at all points
  in the reactor.
• Summing Eq. (22) over all g yields back Eq. (21),
  since

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Will the Precursor Have a Prompt Jump?

• Eq. (18) gave us the general solution for the
  precursor
• The 2nd term will decay away very quickly. Lets
  evaluate the first term, using n1 and ?1 from Eq.
  (19)
• Thus the precursor concentration has a smooth
  exponential behaviour, no prompt jump/drop.

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More on the Prompt Jump/Drop

• The prompt jump or drop holds even if the reactor
  was not initially critical.
• Thus, each time there is a sudden insertion of
  reactivity, there is a step change in reactivity,
  the neutron density (or flux/power) will change
  by a factor ?/(?-?).
• Following the prompt jump/drop, the time
  evolution of the neutron density (or flux or
  power) will be according to the stable period
  1/?1.

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Prompt Criticality

• The condition ? ? corresponds to
• This means that in this case, even if we ignore
  the delayed neutrons (? ), keff will be or gt1,
  i.e., the reactor is critical on prompt neutrons
  alone. This is prompt criticality.
• The delayed neutrons then no longer play a
  crucial role, and when ? increases beyond ?
  (prompt supercriticality), very very short
  reactor periods (lt 1 s, or even much smaller,
  depending on the magnitude of ?) develop.
• Thus, it is advisable to avoid prompt
  criticality.

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• END