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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 < 1.0000 ? subcritical (power decreasing)keff > 1.0000 ? supercritical (power increasing).

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## Nuclear Reactor Kinetics

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**1. **Nuclear Reactor Kinetics Craig Marianno
NE 113

**2. **Effective Multiplication Factor (keff)

**3. **keff and Power Power is directly proportional to neutron density
keff = 1.0000 ? critical (power constant)
keff < 1.0000 ? subcritical (power decreasing)
keff > 1.0000 ? supercritical (power increasing)

**4. **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

**5. **Reactivity When keff is close to 1.0000, ?k and ? and nearly the same.
Example: keff = 0.98

**6. **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.

**7. **Delayed Neutrons

**8. **Delayed Neutrons

**9. **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!

**10. **Prompt Critical

**11. **Prompt Critical

**12. **Reactivity in Dollars From our previous example:

**13. **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)

**14. **Neutron Lifetime

**15. **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

**16. **Effective Neutron Lifetime(delayed neutrons included) ?eff = effective fraction of delayed neutrons
?eff = effective decay constant of precursors
? = reactivity

**17. **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

**18. **Reactor Kinetics Solving:

**19. **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.

**20. **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

**21. **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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