In preceding chapters, the classification of states of a reactor according to the effective multiplication factor – keff was introduced. The effective multiplication factor – keff is a measure of the change in the fission neutron population from one neutron generation to the subsequent generation. But sometimes it is convenient to define the change in the keff alone, the change in the state, from the criticality point of view.
For these purposes reactor physics use a term called reactivity rather than keff to describe the change in the state of the reactor core. The reactivity (ρ or ΔK/K) is defined in terms of keff by the following equation:
From this equation it may be seen that ρ may be positive, zero, or negative. The reactivity describes the deviation of an effective multiplication factor from unity. For critical conditions the reactivity is equal to zero. The larger the absolute value of reactivity in the reactor core, the further the reactor is from criticality. In fact the reactivity may be used as a measure of a reactor’s relative departure from criticality.
It must be noted the reactivity can be calculated also according to the another formula.
This formula is widely used in neutron diffusion or neutron transport codes. The advantage of this reactivity is obvious, it is a measure of a reactor’s relative departure not only from criticality (keff = 1), but it can be related to any sub or supercritical state (ln(k2 / k1)). Another important feature arises from the mathematical properties of logarithm. The logarithm of the division of k2 and k1 is the difference of logarithm of k2 and logarithm of k1. ln(k2 / k1) = ln(k2) – ln(k1). This feature is important in case of addition and subtraction of various reactivity changes.
See more: D.E.Cullen, Ch.J.Clouse, R.Procassini, R.C.Little. Static and Dynamic Criticality: Are They Different?. Lawrence Livermore National Laboratory. UCRL-TR-201506. 11/2003.
Units of Reactivity
Mathematically, reactivity is a dimensionless number, but it can be expressed by various units. The most common units for research reactors are units normalized to the delayed neutron fraction (e.g. cents and dollars), because they exactly express a departure from prompt criticality conditions.
The most common units for power reactors are units of pcm or %ΔK/K. The reason is simple. Units of dollars are difficult to use, because the normalization factor, the effective delayed neutron fraction, significantly changes with the fuel burnup. In LWRs the delayed neutron fraction decreases with fuel burnup (e.g. from βeff = 0.007 at the beginning of the cycle up to βeff = 0.005 at the end of the cycle). This is due to isotopic changes in the fuel. It is simple, fresh uranium fuel contains only 235U as the fissile material, meanwhile during fuel burnup the importance of fission of 239Pu increases (in some cases up to 50%). Since 239Pu produces significantly less delayed neutrons (0.0021 for thermal fission), the resultant core delayed neutron fraction of a multiplying system decreases (it is the weighted average of the constituent delayed neutron fractions).
βcore= ∑ Pi.βi
The unit of reactivity which has been normalized to the delayed neutron fraction. Reactivity in dollars = ρ / βeff. The cent is 1/100 of a dollar. This is very useful unit, because the reactivity in dollars (rather in cents) determines exactly the response of the reactor on the reactivity insertion. Conversion of dollars to pcm depends on βeff. For reactor core with βeff = 0.006 (0.6%) one dollar is equal to about 600 pcm. It is very important amount of reactivity, because if the reactivity of the core is one dollar, the reactor is prompt critical.
BOC and βeff = 0.006
keff = 0.99 ρ = (keff – 1) / keff = -0.01 ρ = -0.01 / 0.006 = -1.67 $ = -167 cents
EOC and βeff = 0.005
keff = 0.99 ρ = (keff – 1) / keff = -0.01 ρ = -0.01 / 0.005 = -2.00 $ = -200 cents
The unit of reactivity in percents of the effective multiplication factor. For example, the subcriticality of keff = 0,98 is equal to -2% in units of %ΔK/K. Since this is very large amount of reactivity, these units are usually used to express significant quantities of reactivity like power defects, xenon worth, integral worth of control rods or shutdown margin. For operational changes that affect the effective multiplication factor this unit is inappropriate, because these changes are of the lower order.
keff = 0.99 ρ = (keff – 1) / keff = -0.01 ρ = -0.01 * 100% = -1 %
percent mille (pcm)
The unit of reactivity which is one-thousandth of a percent %ΔK/K (equal to 10-2x10-3 = 10-5 of keff). The unit of pcm is used at many LWRs because reactivity insertion values are generally quite small and units of pcm allows reactivity to be written in whole numbers. The operational changes such as control rods movement causes usually reactivity insertion of the order of units of pcm per one step. The fact that the effective delayed neutron fraction changes with the fuel burnup have an important consequence. Due to the difference in βeff a response of a reactor on the same reactivity insertion (in units of pcm) is different at the beginning (BOC) and at the end (EOC) of the cycle.
For example, one step of control rods causes greater response at EOC than at BOC. Despite the fact that we assume in both cases, that one step causes the same reactivity insertion (e.g. +10pcm). Moreover, this assumption is not always correct, because the control rods worth increases with fuel burnup.
(10 pcm = 1.43 cents for βeff = 0.007; 10 pcm = 2.00 cents for βeff = 0.005)
keff = 0.99 ρ = (keff – 1) / keff = -0.01 ρ = -0.01 * 105 = -1000 pcm
Reactivity in Reactor Kinetics
In preceding chapters, the classification of states of a reactor according to the effective multiplication factor – keff was introduced. The effective multiplication factor – keff is a measure of the change in the fission neutron population from one neutron generation to the subsequent generation. Also the reactivity as a measure of a reactor’s relative departure from criticality was defined.
In this section, amongst other things it will be briefly described how the neutron flux (i.e. the reactor power) changes if reactivity of a multiplying system is not equal to zero. An understanding of the time-dependent behavior of the neutron population in a nuclear reactor in response to either a planned change in the reactivity of the reactor or to unplanned and abnormal conditions is of the most importance in the nuclear reactor safety. This subject is usually called reactor kinetics (without reactivity feedbacks) or reactor dynamics (with reactivity feedbacks and with spatial effects).
Nuclear reactor kinetics is dealing with transient neutron flux changes resulting from a departure from the critical state, from some reactivity insertion. Such situations arise during operational changes such as control rods motion, environmental changes such as a change in boron concentration, or due to accidental disturbances in the reactor steady-state operation.
Point Kinetics Equation – One Delayed Neutron Group Approximation
The simplest equation governing the neutron kinetics of the system with delayed neutrons is the point kinetics equation. This equation states that the time change of the neutron population is equal to the excess of neutron production (by fission) minus neutron loss by absorption in one mean generation time with delayed neutrons (ld). The role of ld is evident. Longer lifetimes give simply slower responses of multiplying systems. The role of reactivity (keff – 1) is also evident. Higher reactivity gives simply larger response of multiplying system.
If there are neutrons in the system at t=0, that is, if n(0) > 0, the solution of this equation gives the simplest point kinetics equation with delayed neutrons (similarly to the case without delayed neutrons):Let us consider that the mean generation time with delayed neutrons is ~0.085 and k (k∞ – neutron multiplication factor) will be step increased by only 0.01% (i.e. 10pcm or ~1.5 cents), that is k∞=1.0000 will increase to k∞=1.0001.
It must be noted such reactivity insertion (10pcm) is very small in case of LWRs (e.g. one step by control rods). The reactivity insertions of the order of one pcm are for LWRs practically unrealizable. In this case the reactor period will be:
T = ld / (k∞-1) = 0.085 / (1.0001-1) = 850s
This is a very long period. In ~14 minutes the neutron flux (and power) in the reactor would increase by a factor of e = 2.718. This is completely different dimension of the response on reactivity insertion in comparison with the case without presence of delayed neutrons, where the reactor period was 1 second.