The reactor kinetics equations derived in Section 1-3.2 are labeled “point” kinetics since the neutron density, n, was considered only as a function of time. Actually, n should be written as n(x, y,/,t) to emphasize its spatial dependence Similarly, the density of delayed-neutron emitters should be written C,(x, y,z, t), and each of the fission-product concentrations should be written as func­tions of x, y,z, t. The spatial dependence of the neutron flux depends on many’ factors, such as fuel loading, primary — coolant-sy’stem structure, and reactor-vessel penetrations It also depends on the past history of the individual fuel elements

Since the neutron flux in the fuel determines the heat generated, it follows that knowledge of the spatial distribu­tion of the flux is necessary if the reactor is to be used safely and efficiently as a heat source. Instrumentation systems must be included to provide this knowledge to the operator.

If the neutron flux is not constant throughout the reactor, then the kinetics of the chain reaction is not the same in all parts of the reactor The preceding section shows that many reactivity effects are flux-dependent For example, if a reactor is operating with the neutron flux less in one region of the reactor than in another region, then the equilibrium concentrations of the fission-product poisons in the two regions will not be the same, nor (if the reactor has run this way for any length of time) will the fuel burnup be the same in the two regions

Nonuniform spatial distribution of neutron flux can lead to self-induced oscillations of the flux level, the so-called xenon instability or flux tilt The oscillations result from the fact that regions with different neutron-flux levels have different equilibrium 13,Xe concentrations (Fq. 116) If the flux is increased in a region where it has been low, then the increased flux reduces the 13SXe concentration (since more is removed by neutron capture) and, thereby, the poisoning The reduction in 1 35Xe is not offset immediately by 135| decay since the 13SI concentra­tion has been set by the previous (lower) flux level The net result is that the flux tends to keep on increasing On the other hand, m regions where the flux has been high, a decrease in flux tends to increase the 135Xe (fewer art removed by neutron capture), again with no immediate compensation by decreased 135I The tendency of the increasing flux to keep on increasing and the decreasing flux to keep on decreasing is eventually reversed by the 13SI decay’—the peaking of the xenon poisoning shown m Fig 18 The net result is an oscillation in 1 35Xe poisoning between the regions of the reactor with a period given by

2 it

Period of xenon oscillation = 77717 (1 29)

[Xi(Ax + ахФ) l*

where A| = disintegration constant of 135 I (2 9 X 10 s/sec) Ax = disintegration constant of 1 3sXe (2 1 X 10 5/ sec)

ax — microscopic thermal-neutron cross section of 1 3 5 Xe

ф = thermal-neutron flux

If the flux is 5 X 1013 neutrons cm 2 see 1 , the period of the oscillation is 23 hr, somewhat longer than either the 1 3 51 half-life (6 7 hr) or the 1 3 5 Xe half-life (9 2 hr). As the flux is lowered, the period of the xenon oscillation

approaches 70 hr, at 1014 neutrons cm 2 sec 1 , it is about 17 hr.

The possibility of such flux oscillations must be taken into account in designing the reactoi control system ‘1 Im­possibility of high and low regions of heat generation can introduce potential hazards if there is a natural mechanism that makes the high higher and the low lowet Instrumenta­tion must be provided to sense the onset of an such oscillations.

Whether the sensor has readout capability or not, its proper calibration involves subjecting it to precise pressures

CONNECTING TUBING NOTES

A CONNECTING LINE SHOULD BE AS SHORT AS IS PRACTICAL EQUIVALENT VERTICAL HEAD (VERTICAL DISTANCE BETWEEN PRESSURE SOURCE AND TRANSMITTER) IN PSIG MUST BE LESS THAN 20% OF RANGE OF TRANSMITTER

В DO NOT ANCHOR TUBE SO TIGHT THAT IT CANNOT EXPAND DURING BLOWDOWN

C USE PRESSURE SNUBBER FOR ALL WATER FLOW GAS FLOW HIGH PRESSURE (1500 PSIG OR ABOVE) HIGH VELOCITY STEAM FLOW ANY FLOW WHERE A PUMP IS USED OR WHEREVER RAPID PRESSURE OSCILLATIONS ARE ANTICIPATED

ALL SIZES AND MATERIALS LISTED CONFORM TO THE LATEST REVISION OF THE CODE FOR PRESSURE PIPING ASA B31 1 WHERE MATERIALS ARE NOT NOTED OR WHERE DIFFERENT FITTINGS ARE TO BE USED ALWAYS SELECT MATERIALS THAT CONFORM TO SAID CODE DO NOT CHANGE ANY SIZE TO ONE WHICH WILL NOT MEET THE CODE

Fig. 4.22—Recommended connecting tubing or piping for pressure transmitters. (Courtesy Bailey Meter Company)

and reading out on accurate gages. Correct readout volt­meters, ammeters for use with component signal con­ditioners, and other necessary accessories are recommended by the various manufacturers. Equipment for developing pressure varies according to the magnitude of the desired pressure.

For very low pressures, water, oil, and mercury manom­eters can be used. The bore should be large enough to provide an accurate column reading of the deflection scale. Air pressure from a compressor or a vacuum-pump source is also needed for a complete setup.

For medium pressures, transfer gages having calibration traceable to the U. S. Bureau of Standards can be used. Where water is used in the gage, the readings must be corrected for the weight of water unbalance between the sensor and the master gage.

For high pressures, a deadweight tester must be used. These are available for pressures from 15 to 10,000 psi. Constructed of stainless steel, using distilled water, and including a self-contained hand pump, the deadweight tester is an important calibrating device and may be used to calibrate master transfer gages for medium-pressure work.

Noise techniques, the class of reactor dynamics ex­periments in which no external excitation signal is used, are among the categories listed in Table 6 6, namely

1. Vanance-to-mean method (sometimes called the Feynman method)

2 Spectral analysis or the time-domain equivalent, autocorrelation.

3. Cross spectral analysis or the time-domain equivalent, cross correlation.

Other noise-analysis techniques (such as various kinds of probability analysis of individual pulses) are not sufficiently related to transfer functions to warrant discussion here, but they have been treated in Refs. 113, 113a, and 113b along with the three techniques cited above. In power reactors only the last two methods are used, whereas in zero-power reactors all three methods can be used.

Whether reactor dynamics are studied by introducing external excitation or by relying on the reactor’s intrinsic self-induced noise, the data-acquisition and data-processing hardware are almost the same. Noise methods, of course, process no signals from excitation equipment Table 6.11, having much in common with Table 6.10, shows the types of equipment used in the various noise-analysis experiments described briefly here. Most of the equipment is used for spectral and cross spectral analysis, and only that involving the gate scaler or ion-chamber current integrator is used for the variance-to-mean method

In the variance-to-mean method, the dynamic constants in the neutron kinetic equations can be determined by using a digital computer to give _

1 The variance of neutron-detector counts [c2 — (c) 2 ] taken many times over a time interval or “gate,” r.

2. The average count, c, during г

Results for various gate times30 can be shown to conform to the following equation

7

59eE Ъ Go(7j) V

J=1

Fig. 6.8—Results of determinations of the variance-to-mean ratio for many counter gate times on the Ford reactor.1 13

The constants of Eq 6 21 appear in the zero-power transfer function (6 22) and are given in Table 6.12. By fitting Eq. 6.21 to data, as in Fig 6.8, you can evaluate the constants, especially 7i = (0 ~P)/1

Related to this method are others, such as the Mogil’ner method,33 based on the probability of no counts in a time interval, and a count interval-distribution method of Babala 32 These and many similar techniques of time-domain
analysis of neutron pulses in zero-power reactors have been extensively reviewed in Refs 113a and 113b

The constant є in Eq 6.21 is important in noise measurements of zero-power reactors It is the detector efficiency and is defined as

number of counts/sec (6 23)

number of reactor fissions/sec ‘

Evidently є is the probability of detecting an individual fission. Counters are located in or quite near the reactor core to obtain the values above about 10~s which are needed for a successful experiment. For very large reactors only a zone near the detector contributes neutrons and determines an effective efficiency.

Spectral analyses of ion chambers or fluctuations of other variables are usually accomplished experimentally in one of two ways (1) by passing the signal through a narrow band-pass filter tuned sequentially to the various desired frequencies or (2) by obtaining the autocorrelation function of the signal and then performing a Fourier analysis at the various desired frequencies. (Direct Fourier analysis of the signal is rarely done.) Table 6.11 indicates the various combinations of equipment that can be used to accomplish one or the other of these approaches

In spectral analysis of the signal from an ion chamber in a zero-power reactor, the shape of the transfer function, G0, is obtained directly from the measured P(f) of the detection of particles by using the relation11 3

— = 1 + 0.795 elG0(f)l2 (6 24)

eFo

where F0 is the number of reactor fissions per second Again є must exceed about lCf5 for successful experiments For the ion-chamber noise in a power reactor, the spectrum Pn(f) of n(t) in Fig 6 4 is measured, its fluctua­tions being induced by an internal noise source, kln(t) Evidently,

Pn(f)= lkln(f)l2 lG(f)l2 (6 25)

where G(f) is given by Eq 6 20 Thus ion chamber noise analysis in a power reactor gives information about both the transfer function and the input reactivity noise

Figure 6 9 shows typical results for power operation and how the spectrum differs from the zero-power spectrum Figure 6 10 indicates that large pressurized-water reactors of similar structure have similar noise spectrums

In the more informative power reactor experiments, two signals are observed simultaneously and correlated to obtain a transfer function between them One way to do this follows from Eqs 6.19 and 6 20 The terms x(t) and y(t) are any two system variables whose fluctuations are related After their cross correlation function has been measured, their cross spectrum can then be determined An accurate transfer function can be obtained from Eq. 6 20 if x(t) and y(t) depend primarily on the same noise-source excitation, і e, if they have a high coherence, Eq 6 8.

As indicated in Table 6 11, a cross-spectrum analyzer can be used directly, it is not necessary to determine the cross-correlation function first Equation 6.20 is still used to obtain a transfer function As in cross correlation, only two variables at a time are treated in multivariable systems In Table 6.6 a number of cross-correlation and cross­spectrum experiments in zero-power reactors are noted This approach has been used to measure a quantity proportional to just the second term of Eq 6 24 since the cross spectral density of detection events in two ion chambers is70,72

Pxy(f) = 0.7956162F0 lG0(f)l2 (6 26)

where є] and e2 are their efficiencies Although accuracy analysis70 indicates that G0(f) may theoretically be de­termined to the same precision from Eq 6 24 or 6 26 (assuming the same total detection volume, location, and data-collection time for the single detector and the pair of detectors), experimentalists have indicated preference for the method using two detectors