In NUKEFACT #5 we discussed why reactivity (alone) does not determine the direction of power change. Specifically, the reactor rate equation clearly states that there are two factors that contribute to off-critical reactor rate, namely the rate of reactivity change with time, rho-dot, and the value of reactivity, rho, in the form of (lambdaeff x rho). The rho-dot term accounts for an ongoing change in source multiplication with time. And, since source multiplication occurs through the production of prompt neutrons, the rho-dot term represents the prompt neutron contribution to reactor rate. The reactivity term accounts for ongoing change in the neutron source strength with time. And, since the neutron source is delayed neutrons, the rho term, in the form of (lambdaeff x rho), represents the delayed neutron contribution to reactor rate.
Physically, the prompt neutron population may be either increasing or decreasing due to a corresponding change in source multiplication. Likewise, the delayed neutron source strength may be either increasing or decreasing due to a corresponding change in the precursor inventory. If reactivity rate is negative, then an ongoing reduction in source multiplication is occurring, which, by itself, causes power to decrease with time. If reactivity rate is positive, then an ongoing increase in source multiplication is occurring, which tends to increase power with time. If reactivity is negative, then an ongoing reduction in the delayed neutron source strength is underway, which, by itself, causes power to decrease with time. If reactivity is positive, then an ongoing increase in delayed neutron source strength is occurring, which tends to increase power with time.
A most significant feature of the two contributors to reactor rate is that they are independent of each other. Reactivity rate may be positive or negative, regardless of the algebraic sign of reactivity. Physically, this means that one, but not necessarily both contributors, need be acting in the direction of ongoing power change. Hence, if the two contributors to reactor rate exhibit opposite algebraic sign, then it is the contributor of largest magnitude that determines the direction of ongoing power change, i.e. whether power is increasing or decreasing with time. The direction of ongoing power change depends on the nuclear state of the core, a state which is fully defined only when two factors are specified, namely reactivity rate and reactivity.
Now consider the Reactor Rate Diagram.
This diagram is nothing more than a graphical display of the reactor rate equation. It is a set of precalculated reactor rates presented in an organized manner. As explained in the example given in NUKEFACT #8, the method for obtaining reactor rate is to enter the diagram at the specified rho value on the reactivity scale and then move upward to the appropriate curve, namely stable rate, ramp out, or ramp in, each representing a specific value of rho-dot. Thus, in using the Reactor Rate Diagram the nuclear state is defined by two factors, reactivity rate and reactivity, just as it is in the reactor rate equation.
In the Example given in NUKEFACT #8, for the nuclear state defined as rho = +10x10-4 and rho-dot = +2x10-4 delta-rho/sec, the reactor rate read from the Reactor Rate Diagram was SUR = +1.4 DPM, or T = +18 seconds. The reactor was supercritical and the positive reactivity was increasing with time. Both the prompt and delayed components were acting to cause an ongoing increase in power. Now consider a slightly different situation.
Example: given rho = +10x10-4 and rho-dot = 0 delta-rho/sec, find the reactor rate ..... Enter the Reactor Rate Diagram on the reactivity scale at rho = +10x10-4 and move upward to the stable rate curve for rho-dot = 0 delta-rho/sec. Move horizontally to the Startup Rate scale and read SUR = +0.5 DPM. The reactor period is T = 26/0.5 = +52 seconds.
Here, the reactor is supercritical and reactivity is constant, i.e. rho-dot = 0 delta-rho/sec. Only the delayed neutron component acts to cause power increase with time. The ongoing change in the delayed neutron source strength causes power to rise at a SUR of +0.5 DPM. And since this is the same reactivity value as for the first example, the delayed neutron contribution must also be +0.5 DPM in that case. Then, by difference, the contribution of the prompt neutron component in the first example is +1.4 - 0.5 = +0.9 DPM. This contribution is visible on the Rate Diagram as the vertical displacement between the stable rate curve and the ramp-out curve at rho = +10x10-4.
The only difference between the two examples is that in the first a rho-dot existed, while in the second reactivity was constant (rho-dot = 0). The reactivity value in both cases was identical at rho = +10x10-4. In the first example the reactor rate was +1.4 DPM, while significantly lower at +0.5 DPM in the second example. The first example had both contributors to reactor rate producing an ongoing increase in power. The second example had only one of these contributors producing an ongoing increase in power. This makes sense. For a given reactivity value, the reactor rate is greater when positive reactivity is increasing with time, as during control rod withdrawal, than when reactivity is constant. Maybe rho-dot does make a difference