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# Kinetics With Delayed Neutrons - PowerPoint PPT Presentation

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

B. Rouben

McMaster University

EP 4P03/6P03

2008 Jan-Apr

• 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:

•  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

• Delayed neutrons change this significantly.

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

• We can divide by n to get an equation for .

• Eq. (6) is usually recast into another form (the Inhour equation) by substituting

• 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]

• 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:

• 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 .

• 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

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

• 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:

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

The general solution is

• 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 ]

• Illustration of prompt jump – prompt drop is similar:

(Lamarsh Fig. 7.4)

• 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

• 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.

• 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.

• 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.