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Nuclear Reactor Kinetics

- Craig Marianno
- NE 113

Effective Multiplication Factor (keff)

keff determines whether the neutron density

within a reactor will remain constant or change.

keff and Power

- Power is directly proportional to neutron density

- keff 1.0000 ? critical (power constant)
- keff lt 1.0000 ? subcritical (power decreasing)
- keff gt 1.0000 ? supercritical (power increasing)

k Excess

- Any difference between a given value for keff and

1.0000 is called the excess multiplication

factor (?k) - ?k keff - 1.0000 k excess

Reactivity

- When keff is close to 1.0000, ?k and ? and nearly

the same. - Example keff 0.98

Delayed Neutrons

- Single most important characteristic for reactor

control - Delayed neutrons ? decay of fission products

(precursers) - Prompt neutrons ? fission
- Fraction of delayed neutrons ?
- Delayed neutrons are more effective than prompt

because they are born at a somewhat lower

energy.

Delayed Neutrons

Delayed Neutrons

Delayed Neutrons

- While it is true that they are only a small

fraction of the total neutron population, they

play a vital role in reactor kinetics. - Why?
- They significantly increase the neutron cycle

lifetime!

Prompt Critical

If you depend on the fission neutrons ONLY to

produce further fission events, the system is

said to be prompt critical. If it takes only a

fraction of a second for the next generation of

neutrons to be produced, thermalize and produce

another fission, what do you suppose happens?

Prompt Critical

Reactivity in Dollars

- From our previous example

Neutron Lifetime

- For reactor kinetics, it is important to know

the average time elapsing between the release of

a neutron in a fission reaction and its loss from

the system either by absorption of escape. This

is typically called the prompt neutron

lifetime. This time can be divided into - 1) Slowing Down Time
- 2) Thermal Neutron Lifetime (Diffusion Time)

Neutron Lifetime

Neutron Lifetime(infinite medium - prompt only)

- ?a total thermal macroscopic absorption cross

section - ?a absorption mean free path
- v mean velocity (2200 m s-1)
- Note - finite size reduces average lifetime due

to leakage - - ?a for a core includes all materials

Effective Neutron Lifetime(delayed neutrons

included)

- ?eff effective fraction of delayed neutrons
- ?eff effective decay constant of precursors
- ? reactivity

Reactor Kinetics

- We need to construct an expression for the

number of neutrons per second in the reactor

during a given neutron cycle. - We could use
- n
- k
- l

Reactor Kinetics

- Solving

Reactor Period

- To make the previous equation easier, we can

define the reactor period (T) as T l / ?k. - The reactor period represents the length of time

required to change the reactor power by a factor

of e (2.718). This is why it is sometimes

referred to as the e folding time.

Reactor Kinetics(Prompt Example)

- Assuming the following, what is the increase in

power for a ?k 0.0025 (0.357) at the end of

1.0 s? - ?a 13.2 cm
- v 2200 m s-1

Reactor Kinetics(Delayed Example)

- Assuming the following, what is the increase in

power for a ? 0.0025 (0.357) at the end of 1.0

s? - ? 0.0813 s-1 ?eff 0.007
- v 2200 m s-1 l 6.0X10-5 s

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