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 %
The unit of reactivity which is one-thousandth
of a percent %ΔK/K
(equal to 10-2
). 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
The delayed neutron fraction
, is the fraction of delayed neutrons in the core at creation, that is, at high energies
. But in case of thermal reactors the fission
can be initiated mainly by thermal neutron
. Thermal neutrons are of practical interest in study of thermal reactor behaviour. The effective delayed neutron fraction
, usually referred to as βeff
, is the same fraction at thermal energies.
The effective delayed neutron fraction reflects the ability of the reactor to thermalize and utilize each neutron produced. The β is not the same as the βeff due to the fact delayed neutrons do not have the same properties as prompt neutrons released directly from fission. In general, delayed neutrons have lower energies than prompt neutrons. Prompt neutrons have initial energy between 1 MeV and 10 MeV, with an average energy of 2 MeV. Delayed neutrons have initial energy between 0.3 and 0.9 MeV with an average energy of 0.4 MeV.
Therefore in thermal reactors a delayed neutron traverses a smaller energy range to become thermal and it is also less likely to be lost by leakage or by parasitic absorption than is the 2 MeV prompt neutron. On the other hand, delayed neutrons are also less likely to cause fast fission, because their average energy is less than the minimum required for fast fission to occur.
These two effects (lower fast fission factor and higher fast non-leakage probability for delayed neutrons) tend to counteract each other and forms a term called the importance factor (I). The importance factor relates the average delayed neutron fraction to the effective delayed neutron fraction. As a result, the effective delayed neutron fraction is the product of the average delayed neutron fraction and the importance factor.
βeff = β . I
The delayed and prompt neutrons have a difference in their effectiveness in producing a subsequent fission event. Since the energy distribution of the delayed neutrons differs also from group to group, the different groups of delayed neutrons will also have a different effectiveness. Moreover, a nuclear reactor contains a mixture of fissionable isotopes. Therefore, in some cases, the importance factor is insufficient and an importance function must be defined.
In a small thermal reactor with highly enriched fuel, the increase in fast non-leakage probability will dominate the decrease in the fast fission factor, and the importance factor will be greater than one.
In a large thermal reactor with low enriched fuel, the decrease in the fast fission factor will dominate the increase in the fast non-leakage probability and the importance factor will be less than one (about 0.97 for a commercial PWR).
In large fast reactors, the decrease in the fast fission factor will also dominate the increase in the fast non-leakage probability and the βeff is less than β by about 10%.
Table of main kinetic parameters.
Despite the fact the number of delayed neutrons
per fission neutron is quite small (typically below 1%)
and thus does not contribute significantly to the power generation, they play a crucial role in the reactor control
and are essential from the point of view of reactor kinetics and reactor safety
. Their presence completely changes the dynamic time response
of a reactor
to some reactivity change, making it controllable by control systems such as the control rods
Delayed neutrons allow to operate a reactor in a prompt subcritical, delayed critical condition. All power reactors are designed to operate in a delayed critical conditions and are provided with safety systems to prevent them from ever achieving prompt criticality.
For typical PWRs, the prompt criticality occurs after positive reactivity insertion of βeff (i.e. keff ≈ 1.006 or ρ = +600 pcm). In power reactors such a reactivity insertion is practically impossible to insert (in case of normal and abnormal operation), especially when a reactor is in power operation mode and a reactivity insertion causes a heating of a reactor core. Due to the presence of reactivity feedbacks the positive reactivity insertion is counterbalanced by the negative reactivity from moderator and fuel temperature coefficients. The presence of delayed neutrons is of importance also from this point of view, because they provide time also to reactivity feedbacks to react on undesirable reactivity insertion.