NUKEFACT #28

THE MINIMUM P0SSIBLE STARTUP POWER LEVEL AT RHO = 0

last update July 15, 1997

A question raised frequently by students of reactor behavior is: "During a reactor startup, what is the minimum possible power level when achieving rho = 0 ?" Even though this happens to be a legitimate question, and an understandable curiosity, have you ever seen the answer anywhere? Since the answer relates to "prompt jump", which was discussed in NUKEFACT #27, this seems like a timely subject for this essay.

Before delving into this matter, a review of the magnitude of startup power levels is in order. NUKEFACT #3 introduced the Power Diagram which illustrated equilibrium subcritical power levels over a reactivity range extending from rho = -0.0300 to rho = -.0005. That Power Diagram is reproduced here as Figure 28.1.

FIGURE 28.1 -- THE POWER DIAGRAM

The vertical (log) scale is fission power in watts, extending from 0.01 watts to 10,000 megawatts. The horizontal (linear) scale is reactivity, extending from -0.0300 to +0.0100. The upward sweeping (yellow) curve, starting in the lower left corner of the diagram and extending to criticality is the equilibrium subcritical multiplication curve, expressed as reactor power.

To say the least, the equilibrium subcritical power curve lies in a range of extremely low power. At shutdown, with rho = -0.0300, and a non-fission source strength of 1x10⁸ neutrons/second, power is less than 0.10 watts. This power is easily calculated, as follows:

28.1

Near criticality, with increased source multiplication, the equilibrium power curve lies in the range of 1 to 10 watts, still extemely low.

Since the question of minimum power at criticality involves a transient situation that traverses the Sub-Critical region, the equilibrium power curve, or equilibrium subcritical multiplication curve, provides a useful line of reference. Each point on this curve represents a steady state condition where the emission of source neutrons just balances the loss of fission neutrons (due to k_eff being < 1.0000) to create an equilibrium condition. If, as during the process of reactor shutdown, power is greater than equilibrium for a given constant subcritical reactivity, power will decay until it reaches the equilibrium point. If, as during the process of reactor startup, power is less than equilibrium for a given constant subcritical reactivity, power will increase until it reaches the equilibrium point. When subcritical at a specific reactivity, and with both reactivity and the non-fission source constant, there is only one possible equilibrium power condition, and that lies on the equilibrium subcritical multiplication curve. A single, unique, equilibrium power level exists for each constant subcritical reactivity condition.

So, how does reactor power respond relative the the equilibrium subcritical multiplication curve during a normal startup? In this case let's assume that reactivity is introduced as a ramp, i.e. is added at a constant reactivity rate (rho-dot), from shutdown to rho = 0, without any of the normal interruptions associated with an operational startup. If the reactivity rate is small, e.g. +0.0001 delta-rho/second, power will initially track the equilibrium curve and will then begin to fall slightly below the equilibrium curve as rho = 0 is approached. The power trace for a ramp rate of +0.0001 delta-rho/second is the blue curve on Figure 28.2, shown below. The displacement of the startup curve below the equilibrium curve is small. The reason that startup power falls below the equilibrium curve is that in the transient situation the delayed neutrons lag behind as source multiplication increases. During continuous reactivity addition from shutdown, the startup power will never exceed the equilibrium subcritical multiplication power.

FIGURE 28.2 -- POWER AT RHO = 0 for VARIOUS RAMP RATES

For a more rapid reactivity ramp, the startup power trace falls further below the equilibrium subcritical power curve, because the delayed neutrons lag further behind in the transient. The red trace on Figure 28.2 is for a reactivity ramp rate of +0.0003 delta-rho/second. In the limit, for the most rapid change in reactivity, a step change, the power change is by prompt jump and the power on reaching rho = 0 is at an absolute minimum, albeit greater than the shutdown power. The prompt jump change in power is due solely to the increase in the source multiplication factor, 1/(beta - rho). The green trace on Figure 28.2 is the prompt jump power transient for a step change in reactivity to rho = 0. Thus, it can be seen that as the reactivity ramp rate is increased, the power level on reaching rho = 0 is lower the greater the ramp rate, and naturally is at its lowest possible value when the ramp rate is infinite, as for the step change in reactivity.

To determine this minimum power, it is first necessary to calculate the Prompt Jump Factor (PJF), which is:

28.2

Then, applying the Prompt Jump Factor to the initial power, the core power at criticality is:

28.3

Thus, the means for attaining the miniumum power at rho = 0 is by a hypothetical step change in reactivity, from shutdown to exactly rho = 0. Of course, the absolute power at rho = 0 depends on the shutdown power level, which in turn depends on the non-fission neutron source strength and on the shutdown reactivity.

• Non-fission source strength: If, in the above example, the non-fission source were 10 times stronger, i.e. 1x10⁹ neutrons/second, then the shutdown power would be 10 times greater, or 0.44 watts. The equilibrium multiplication curve would be translated upward on the Power Diagram by one decade. And, since for the same shutdown reactivity, the PJF would not change, the minimum power at rho = 0 would be 10 times greater, or 2.46 watts.

• Shutdown reactivity: If, in the above example, the shutdown reactivity were rho = -0.1000, then the shutdown power would be lower, at 0.013 watts, assuming the non-fission source remains at 1x10⁸ neutrons/second. But the Prompt Jump Factor to rho = 0 would then be 16.38 and the prompt jump power at rho = 0 would be 0.214 watts.

And, to extend the question, what is the maximum possible startup power at rho = 0 ? Simply put, it is defined by the (yellow) equilibrium source multiplication curve on Figure 28.2. Whereas, the minimum possible startup power occurs with the fastest possible reactivity ramp, which is a step, the maximum possible startup power occurs with the slowest possible reactivity ramp, which is one where the rate approaches 0 delta-rho/second. The maximum possible startup power is essentially the equilibrium power immediately before attaining criticality. The only question is, "what value of reactivity is to be used to represent immediately before criticality". As a practical matter, we think rho = -0.0001 is representative. (If you like something slighty closer to criticality, we won't argue the point.) The maximum power is then:

28.4

SUMMARY

The startup power at rho = 0 is bounded by the equilibrium power curve, which acts as an upper limit, or maximum power at rho = 0, and by the step change curve, which acts as a lower limit, or minimum power at rho = 0. For the examples given, the minimum possible startup power at rho = 0 is 0.246 watts and the maximum possible startup power at rho = 0 is 13.1 watts. For an actual startup, at some intermediate reactivity rate, which is never known exactly by the Reactor Operator, the startup power level at rho = 0 would lie in this range of power, i.e. between 0.246 and 13.1 watts.

Prompt jump and prompt drop can be introduced, and practiced, in the class room on THE REACTOR TRAINER, as can investigation of the effect of reactivity rate on the startup power at rho = 0.