## Inverse Reactor Kinetics – Reactimeter

The reactivity describes the measure of a **reactor’s relative departure from criticality**. During reactor operation and during reactor startup, it is important to monitor the reactivity of the system. It must be noted, **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 **startup rate** (especially in case of PWRs).

On the other hand, during reload startup physics tests which are performed at the startup after refueling the commercial PWRs, it is important to monitor subcriticality continuously during criticality approach. On-line reactivity measurements are based on the inverse kinetics method. The inverse kinetics method is a reactivity measurement based on the point reactor kinetics equations. This method can used for:

**Reactivity measurement at high neutron level**–**reactimeter without source term**. Without source term a reactimeter can be constructed, but it works only at higher neutron levels, where the neutron source term in point reactor kinetics equations may be neglected.**Reactivity measurement at subcritical multiplication**–**reactimeter with source term**. For operation at low power levels or in the sub-critical domain (e.g. during criticality approach), the contribution of the neutron source must be taken into account and this implies the knowledge of a quantity proportional to the source strength, and then it should be determined. The subcritical reactimeter is based on the determination of the**source term**(source strength).

As was written, the reactivity of the system can be measured by a **reactimeter**. The reactimeter is a device (or rather a **computational algorithm**) that can continuously give real time reactivity using the **inverse kinetics method**. The reactimeter usually processes the signal from source range excore neutron detectors and calculates the reactivity of the system.

It was shown that the source term is not so easy to be determined and the problem is that the source term is of the highest importance in the subcritical domain. One of the recognize methods for source term determining is known as Least Squares Inverse Kinetics Method (**LSIKM**).

Special reference: Seiji TAMURA, “Signal Fluctuation and Neutron Source in Inverse Kinetics Method for Reactivity Measurement in the Sub-critical Domain,” J. Nucl. Sci. Technol, Vol.40, No. 3, p. 153–157 (March 2003)

## Inverse Kinetics – Equations

The **exact point kinetics equations**, that can be derived from the general neutron balance equations without making any approximations are:

where:

- n(t) is the neutron density in the core (which is proportional to the detector count rate)
- C
_{i}(t) is the precursors density of delayed neutrons of group i - Λ is the prompt generation time
- ρ is the reactivity
- the constants β
_{i}and λ_{i}are the fraction and decay constant of delayed neutron precursor of group i - S(t) is the source term, which characterizes number of neutrons (source neutrons) added to the system from an external source. As can be seen, the source term does not influence the dynamics of the system, since it does not influence the reactivity of the system.

According to Seiji TAMURA, the Inverse Kinetics equations (flux ⟶ reactivity) for discrete time series data can be derived from the ordinary point kinetics equations (reactivity ⟶ flux) assuming that the reactor power change for the interval of Δt is as n(t)=n_{j-1}exp(μ_{j}t), where μ_{j} = log(n_{j}/n_{j-1})/Δt:

When the reactor power is at a steady state n_{o}, there are seven initial conditions that can be obtained as:

The equation for ρ_{j} provides a real time reactivity calculation for successive reactor power data. When the reactor is operating at sufficiently high power level the last term of the right side, the source term, may be neglected, because it becomes negligible.

For operation at low power levels or in the sub-critical domain (e.g. during criticality approach), the contribution of the neutron source must be taken into account and this implies the knowledge of a quantity proportional to the source strength, and then it should be determined. Otherwise zero reactivity will be obtained for any subcritical reactor condition (after the transition phenomena have died out). The source term can be determined using data from a transient state after the introduction of a given reactivity. The source term can be determined from data from rod drop experiment using the Least Squares Inverse Kinetics Method (LSIKM).

**Special reference:** M. Itagaki, A. Kitano, “Revised source strength estimation for inverse neutron kinetics,” Preprints 1999 Fall Mtg., At. Energy Soc. Jpn., Hiroshima, Japan, G20, (1999)

**Special reference:** Renato Yoichi Ribeiro Kuramoto, Anselmo Ferreira Miranda. SUBCRITICAL REACTIVITY MEASUREMENTS AT ANGRA 1 NUCLEAR POWER PLANT. 2011 International Nuclear Atlantic Conference – INAC 2011. ISBN: 978-85-99141-04-5.