When first developing the concept of keff and its effect on reactor behavior with time, the following equation is usually employed:
This equation is based upon an average fission neutron, one whose lifetime, called a generation time, is the weighted average of the prompt neutron and delayed neutron lifetimes. It is used to demonstrate the following relationship between any constant keff and neutron behavior with time.
........ keff .............. Neutron Population ................Condition
.. > 1.0000 .......... exponentially increasing ..........supercritical
.. = 1.0000 .......... constant ....................................critical
.. < 1.0000 .......... exponentially decreasing ........ subcritical
Ultimately, a more refined single delay group model is developed from the neutron balance equation which provides an excellent basis for understanding and representing reactor behavior with time, i.e. by treating the prompt and delayed neutrons separately. But before proceeding to that definition, it is helpful to review the general equation for reactor power, which is:
This equation applies to any operational power condition, steady state or transient, from shutdown to full power. It reveals that the only two possible causes of power change with time (on the left-hand-side) are ongoing change in the precursor inventory, C, or ongoing change in rho, or keff , (on the right-hand-side).
For the single delay group transient situation, without an extraneous neutron source, power is represented as:
On the right hand side of this equation, the portion of the exponent in square brackets is the reactor rate, which is a measure of the rate of power change with time. In the U.S., PWRs express reactor rate as startup rate while BWRs use reactor period, but more on that in the future. The reactor rate can be factored, as follows:
The two terms on the right-hand-side of this equation indicate that there are two contributors to reactor rate. The first term accounts for ongoing change in keff with time. It can be shown that this term is directly related to the prompt neutrons that "multiply" the delayed neutron source. The second term accounts for ongoing change in the delayed neutron precursor emissions. Note that these two terms correspond to the two parameters identified in the general equation for power as being the only two possible causes of power change with time.
Reactor rate is positive for power increase with time and negative for power decrease with time. Reactivity rate, which determines the algebraic sign of the prompt neutron term, may be either positive, as for ongoing control rod withdrawal, or negative, as for ongoing control rod insertion. Reactivity, which determines the algebraic sign of the delayed neutron term, may be either positive, as for a supercritical condition, or negative, as for a subcritical condition. The algebraic signs of the two contributors to reactor rate can be in opposition, which means that the term of greatest magnitude determines whether power is increasing or decreasing. It is not uncommon for reactivity-rate to be the dominant factor causing power change.
Thus, reactivity alone does not determine the direction of power change. The direction of power change is dependent on the net effect of two factors, reactivity and reactivity rate. Only for the special case of constant reactivity, rho-dot = 0, is ongoing power change dependent on reactivity alone.
And, lest you be inclined to dismiss this as being merely of academic interest, realize that this leaves the reactor operator with an impossible task. He is expected to know the nuclear status of the core, whether subcritical or supercritical, based on reactor rate (or the direction of power change). It can't be done with certainty. That is why reactivity computers were invented.
The negative reactivity rate is crucial during automatic reactor scram to terminate a power excursion. The Chernobyl operators apparently failed to understand this, which contributed to the devastating results of that accident. A rather expensive lesson. Power reversal during reactor scram will be discussed in a future NUKEFACT.