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## PowerPoint Slideshow about 'Kinetics With Delayed Neutrons' - elia

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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

- 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

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

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

- 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]

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:

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 (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

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

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 (cont.)

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

The general solution is

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 ]

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?

- 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

- 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

- 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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