Page 182 - Dynamics and Control of Nuclear Reactors

13.8 BWR stability problem and impact on control 179

where
G c ¼closed-loop transfer function (feedback effects included).
G o ¼open-loop transfer function (the zero-power transfer function).
H¼feedback (cents/% power).
At low frequencies, the magnitude of G o is large. Consequently, G c ¼1/H at low fre-
quencies. The feedback frequency response can be calculated using
1 1
H ¼ (13.3)
G c G o
The resulting feedback frequency response appears in Fig. 13.7. Note that the phase
lag is more than 90deg. at frequencies above around 0.1rad/s. As shown in
Section 3.8 such phase shifts in system feedback can cause instability if the feedback
gain is large enough.
The frequency response for various feedback conditions can be deduced by
applying a multiplicative factor, K to the feedback term (G o H in Eq. (13.2)).
G o
G c ¼ (13.4)
1+ KG o H
Note that K¼1 for the original low-order model. Fig. 13.8 shows the closed-loop
gain for various values of K. Clearly the resonance at around 0.3rad/s grows as K
increases and shifts to higher frequencies. Ref. [4] shows that the system becomes
unstable at a value of K 2.25.
The above discussion reveals that useful insights can be deduced if basic princi-
ples of dynamic analysis of feedback systems are understood and employed.

13.8 BWR stability problem and impact on control
BWRs exhibit instability at conditions of high power and low recirculation flow.
This instability is caused by a complex coupling of neutronics and thermal-
hydraulics. The basic cause of BWR instability is time lagged flow and reactivity
feedbacks. Recall that positive feedbacks usually cause stability problems, but neg-
ative feedbacks also can cause instability if their effect is delayed (and the feedback
gain is large enough). See Section 3.8.
Subcooled water enters a BWR channel at the bottom. As it flows upward, boiling
occurs and the void fraction increases. A disturbance (typically inlet flow change,
inlet subcooling change, or power change) causes a localized change in steam bubble
concentration in the lower portion of the channel. The propagation of this bubble
packet as it travels up the channel is called a density wave [2]. This density wave
causes changes in the local pressure drops as it propagates.
Consider a disturbance that causes an increase in steam bubble formation in the
lower portion of the channel. The resulting density wave travels upward, and it travels
faster than the flow prior to the disturbance. As a result, the total channel pressure drop