Page 289 - Dynamics and Control of Nuclear Reactors

APPENDIX E Frequency response analysis of linear systems 291

E.4 Systems with time delay dynamics
Many dynamic systems have an inherent time delay. For example, in pipe flow,
a downstream point experiences a disturbance at some time later than an upstream
disturbance. In a nuclear power plant, the hot water from the vessel in a pressurized
water reactor is carried through by hot leg piping and delivered to steam generators.
When a temperature change takes place in the water leaving the vessel, this
change is carried through the flowing water in the hot leg. Depending on the
length of the hot leg, a certain transport time is needed to sense this change at some
other point.
This time delay or transport delay (dead time) between two variables δx(t) and
δy(t) is expressed as

δytðÞ ¼ cδxt DÞ,c isa constant parameter: (E.28)
ð
y(t) detects the changes in x(t) after D seconds. The transfer function of a pure time
delay is

δYsðÞ sD
GsðÞ ¼ ¼ ce (E.29)
δXsðÞ
The frequency response function is given by

ð
GjωÞ ¼ c exp jωDÞ (E.30)
ð
Thus jG(jω)j¼ constant for all ω and the phase angle φ (ω)¼ -ωD; D is the slope of
the linear phase angle plot.

Example E.5
An example of the use of pure delay dynamics between two detector signals is the measurement of
flow velocity in a BWR. The phase angle between two in-core detectors, placed parallel to the core
axis, has a linear form, and the slope of the phase angle corresponds to the time delay in the flow
passage from the upstream detector to the downstream detector.
Fig. E.8 is the core map of a typical BWR, showing the location of in-core neutron detector
strings. The detectors in each string are labeled A, B, C, and D from bottom (upstream) to top
(downstream) of core. Fig. E.9 is a plot of the phase angle between the signals from detectors
B and C. As seen in this plot, the phase angle between the detector signals B and C is linear, indi-
cating that the relationship between these two detector signals may be approximated by a pure time
delay in the frequency band of interest. Properties such as these, including the characteristic fre-
quency response of neutron detectors in BWRs, are used to monitor reactor performance (stability,
flow pattern, etc.) in an on-line manner.
Continued