# Reactor Period

## Reactor Period

Reactivity is not directly measurable and therefore most power reactors procedures do not refer to it and most technical specifications do not limit it. Instead, they specify a limiting rate of neutron power rise (measured by excore detectors), commonly called a reactor period (especially in case of BWRs).

The reactor period, τe, or e-folding time, is defined as the time required for the neutron density to change by a factor e = 2.718. The reactor period is usually expressed in units of seconds or minutes.

At this time: where:
n(t) = transient reactor power
n(0) = initial reactor power
τe = reactor period

The smaller the value of τe, the more rapid the change in reactor power. The reactor period may be positive or negative. If the reactor period is positive, reactor power is increasing. If the reactor period is negative, reactor power is decreasing. If the reactor period is constant with time, as associated with exponential power change, the rate is referred to as a stable reactor period. If the reactor period is not constant but is changing with time, as for non-exponential power change, the period is referred to as a transient reactor period.

Derivation of the formula τe = ld / (k-1) is based on many assumptions and it is only simplest approximation of the reactor period. A much more exact formula reactor period is based on solutions of six-group point kinetics equations. From these equation an equation called the inhour equation (which comes from inverse hour, when it was used as a unit of reactivity that corresponded to e-fold neutron density change during one hour) may be derived. where:
l = prompt neutron lifetime
βeff = effective delayed neutron fraction
λeff = effective delayed neutron precursor decay constant
τe = reactor period
ρ = reactivity The presence of delayed neutrons brings many interesting phenomena. This chart shows the typical response of reactor on constant reactivity insertion.

The first term in this formula is the prompt term and it causes that the
positive reactivity insertion is followed immediately by a immediate power increase called the prompt jump. This power increase occurs because the rate of production of prompt neutrons
changes immediately as the reactivity is inserted. After the prompt jump, the rate of change of power cannot increase any more rapidly than the built-in time delay the precursor half-lives allow. Therefore the second term in this formula is called the delayed term. The presence of delayed neutrons causes the power rise to be controllable and the reactor can be controlled by control rods or another reactivity control mechanism.

The relationship between reactor period and startup rate is given by following equations: Example:

Suppose keff = 1.0005 in a reactor with a generation time ld = 0.01s. For this state calculate the reactor period – τe, doubling time – DT and the startup rate (SUR).

ρ = 1.0005 – 1 / 1.0005 = 50 pcm

τe = ld / k-1 = 0.1 / 0.0005 = 200 s

DT = τe . ln2 = 139 s

SUR = 26.06 / 200 = 0.13 dpm

References:
Nuclear and Reactor Physics:
1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
5. W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
6. G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
7. Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
8. U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.