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Kinetics With Delayed Neutrons. B. Rouben McMaster University EP 4P03/6P03 2008 Jan-Apr. Point-Kinetics Equations.

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Kinetics with delayed neutrons l.jpg

Kinetics With Delayed Neutrons

B. Rouben

McMaster University

EP 4P03/6P03

2008 Jan-Apr


Point kinetics equations l.jpg
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:


Case without delayed neutrons l.jpg
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


Case with delayed neutrons l.jpg
Case with Delayed Neutrons

  • Delayed neutrons change this significantly.

  • To solve the point-kinetics equations, we can try exponential solutions of the form


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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


Inhour equation l.jpg
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]


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General Solution

  • There are (G+2) branches in the graph of the Inhour equation, and (G+1) roots for  (intersections with line ).

  • If  < 0 all roots will be negative, but

  • If  > 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 (G+1) 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 <0), 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 .


General solution cont9 l.jpg
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


Solution for 1 delayed neutron precursor group l.jpg
Solution for 1 Delayed-Neutron-Precursor Group

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


Solution for 1 delayed neutron precursor group cont l.jpg
Solution for 1 Delayed-Neutron-Precursor Group (cont.)

  • Now there are 3 branches and 2 roots for :

    • When  < 0, both  values are negative

    • When  > 0, one  value is positive, and the other is negative.

    • When  > 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:



Solution for 1 delayed neutron precursor group cont13 l.jpg
Solution for 1 Delayed-Neutron-Precursor Group (cont.)

  • If we substitute the form into the point-kinetics equations, we get

    The general solution is


Solution for 1 delayed neutron precursor group cont14 l.jpg
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 ]


Prompt jump or drop l.jpg
Prompt Jump or Drop

  • Illustration of prompt jump – prompt drop is similar:

    (Lamarsh Fig. 7.4)


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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


Will the precursor have a prompt jump l.jpg
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. Let’s evaluate the first term, using n1 and 1 from Eq. (19)

  • Thus the precursor concentration has a smooth exponential behaviour, no prompt jump/drop.


More on the prompt jump drop l.jpg
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.


Prompt criticality l.jpg
Prompt Criticality

  • The condition  =  corresponds to:

  • This means that in this case, even if we ignore the delayed neutrons ( ), keff will be = or >1, i.e., the reactor is critical on prompt neutrons alone. This isprompt criticality.

  • The delayed neutrons then no longer play a crucial role, and when  increases beyond (prompt supercriticality),very very short reactor periods (< 1 s, or even much smaller, depending on the magnitude of ) develop.

  • Thus, it is advisable to avoid prompt criticality.



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