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Kinetics With Delayed Neutrons

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

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  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

  6. 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 (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:

  8. 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 .

  9. 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

  10. 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  < 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:

  12. 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

  14. 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 ]

  15. 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

  17. 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.

  18. 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.

  19. 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.

  20. END

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