Inputs other than control-rod perturbations are considered briefly in this section. The brevity is necessary because there is little experience with tests of this type. Nevertheless, in the long run, results from these tests may be much more useful than reactivity perturbation tests for many systems.

For reactors with separate steam generators (this includes all current power reactors except boiling-water reactors), the overall plant performance is strongly affected by steam generator characteristics. The dynamic perform­ance of the steam generator is influenced by its design (recirculation type or once-through), its size, and its controllers. Also, the heat-transfer and flow characteristics of steam generators can change significantly between cleanings because of crud deposits. Dynamics tests may be useful in assessing the consequences of these changes on system performance. Steam generators can be perturbed by changes in primary fluid temperature or flow, changes in feedwater flow, or by changes in steam flow. Changes in primary fluid temperature can be induced by reactivity perturbations, but these tempera­ture variations may be too small to give suitably large changes in steam generator conditions. In this case, feedwater flow or steam flow perturbations will be necessary to obtain steam generator test data. A possible exception is a once-through steam generator with superheat. These have rather large responses to primary fluid temperature.

Boiling-water reactors are even more strongly coupled to steam conditions than other reactor types. Steam flow and feedwater flow perturbations appear well suited as excitations for these systems. Also, the recirculation flow is readily modulated because it is varied for control purposes during normal operation.

Many of the early tests on reactor systems have focused on a single input-output pair, usually reactivity-neutron flux. However, if a number of important process variables are stimulated strongly enough by the input in a test, they can be analyzed to give frequency response results for each output relative to the selected input. These results may greatly increase the information available from the test.

A key factor in determining whether a particular process variable can be used is the sensitivity and response time of the plant sensor. For many plant sensors, the main duty is monitoring steady-state or slowly varying parameters. The tester must determine whether the sensor sensitivity is large enough to give suitably large signals relative to background noise for expected variations in the measured parameter. The response time is often longer than one might expect because the hostile environment in power reactors requires sturdy equipment, and this often leads to some isolation of the sensor from the process. Good examples are thermocouples and resistance thermometers used for fluid-temperature measurement. They are usually imbedded in a ceramic such as magnesium oxide and cased in a stainless steel sheath. A typical time constant is 3 sec.

In analogy with the power dissipated in a resistor in an electric circuit (P = I2R), the generalized power in any kind of signal is given by

P = x2(t)

signal. The relation

/* T/2 00

xt)dt=T £ C, C_, (2.6.11)

4-772 k= — 00

is called Parseval’s relation for periodic signals.

Cross spectra are defined for pairs of signals. Two related periodic signals with the same period may be used to give

№12)0 = TC0D0, (E2)k ~ 2TCkD_k for к # 0 (2.6.12)

and

(P12)о = C0D0, (P12)k = 2CkD„k for к Ф 0 (2.6.13)

where (£12)* is the component of the cross-energy spectrum at harmonic k, (ЛА the component of the cross-power spectrum at harmonic k, Ck the Fourier coefficient of signal 1 at harmonic k, and Dk the Fourier coefficient of signal 2 at harmonic k. The power spectrum of a single signal is always real. The cross-power spectrum of two different signals may be complex.

Similar results may be obtained for nonperiodic signals. Let y(t) be a general, nonperiodic signal. Then

/* 00

y(t) = (1/2я) Ytiabe^dto (2.6.14)

v — 00

and

Interchanging the order of integration gives

or

E = (1/2я) Y(jco) Y( —jco) dco (2.6.16)

** — GO

This expression gives the total energy in the signal. Since this is obtained by integration over frequency, (jln)Y{j(a)Y(—jw) is interpreted as an energy density (energy per unit frequency). Thus, in a nonperiodic signal, there is a continuous energy density spectrum. The relation

Ґ y2(t)dt = (1/2тг) f°° Y(jco)Y(-jco)dco (2.6.17)

J — 00 v — 00

is called Parseval’s relation for nonperiodic signals.

The block diagram for a power reactor appears in Fig. 5.2. The closed — loop transfer function is

G = GJ{ 1 + GjH)

Fig. 5.2. Power reactor block diagram where Gj(s) = <5n(s)/<5p(s) is the zero-power transfer function and H(s) = <5p(s)/<5n(s) the feedback transfer function.

or

G = 1/[(1/Gj) + H] (5.2.2)

From this, we can conclude

G = Gj when 11/G j I » |Я|

G = 1 /Н when 11/G! I « |H|

The effect of feedback on the closed-loop frequency response can be evaluated using a typical feedback transfer function and the zero-power transfer function (Eq. 5.1.3). The simplest form that might be used to model a feedback transfer function is a first-order lag.

H = Sp/Sn = A/(rs + 1) (5.2.3)

More detailed models would have a number of poles and zeroes, but there will always be a high-frequency break followed by a monotonic decrease in gain with increasing frequency. Typical values for this break are 1 rad/sec for light-water reactors and 10 rad/sec for liquid-metal-cooled fast reactors. (This difference is because fast-reactor fuel elements have smaller diameters and higher surface heat-transfer coefficients.)

Figure 5.3 shows gain plots for a zero-power frequency response and a typical feedback frequency response based on a first order lag. Figure 5.4 shows the closed-loop frequency response for this system. At low frequencies, ll/GJ « H, and G = 1 /Н. At high frequencies, |1 /GJ » |H|, and G ~ Gj. Figure 5.5 shows the effect of the break frequency for a first-order-lag feedback model on closed-loop system gain.

Of course, the feedback reactivity can have phase shifts that cause it to increase the net reactivity over the input reactivity at some frequencies.

c

в

(3

.1 1
Frequency (radians/sec.)

Q.

■o (3 .001

.001 .01 .1 1 10 100
Frequency (radians/sec.)

Fig. 5.3. (a) Amplitude for a zero-power reactor; (b) amplitude for a simple reactivity feedback model.

In this case the gain of the closed-loop frequency response is greater than the gain of the zero-power frequency response at these frequencies. This indicates that the feedback has a destabilizing effect. This often occurs in actual systems, and frequency response tests may be used to monitor this effect with changing system conditions (such as power level). Figure 5.6 shows closed-loop results for the molten-salt reactor experiment at several power levels(2). Clearly, the feedback causes the gain at some power levels to be

larger than the gain at zero power. Furthermore, stability for this system improves with increasing power level.

The main reason for interest in the PRTS for frequency response measure­ments is that the signal discriminates against nonlinear effects (it is anti­symmetric). Because several binary signals also have this property, they are preferred over the PRTS for future applications since the practical problems of inputting a three-level signal into a system are often significantly greater than problems with a two-level signal. Nevertheless, there has been some interest in the PRTS signal, and a discussion of its characteristics is included here.

f See the literature (32, 38, 49, 50, 56-61).

2. The power spectrum becomes flatter as Z increases, as with the PRBS and n sequence.

3. The absolute magnitude of the harmonics decreases as Z increases, as with the PRBS and n sequence. Again, the same bandwidth relations (see Section 2.12) apply.

4. The PRTS is antisymmetric.

The PRTS can be generated using a digital shift register with modulo-3 adder feedback. Signals can be generated for Z = 3" — 1 for integer values of n. Table 3.3 gives feedback connections that provide PRTS signals. Figure 3.10 shows two cycles of a PRTS signal (Z = 8). It should be noted

that while construction of a PRBS generator is simple and requires only the use of standard digital logic components, no comparable components are available for ternary logic. Thus the algorithm for PRTS generation is useful for implementation with a digital computer, but not as a basis for construct­ing a simple signal generator.

To be suitable for a dynamics test, a control rod must be capable of introducing sufficient reactivity at a suitably fast rate. An estimate of the required magnitude of the reactivity perturbation may be obtained from

the zero-power transfer function for one group of delayed neutrons:

n0dp As(s + P/A)

Feedback effects are usually important for w < P/А. The plateau in the frequency response between w = X and w = /?/Л has a magnitude

dn/n0dp = /P (7.1.2)

In individual cases, the full-power frequency response gain may be smaller or larger than this, but this value is useful for making a rough estimate of the reactivity perturbation needed in the test. Solving for dp (expressed in cents), we obtain

dp (in cents) = (dn/n0) x 100 (7.1.3)

Thus a reactivity perturbation of ~ 1 cent would be used to obtain a power swing of ~ 1 %. The reactivity required in a test will depend on the back­ground noise level and the length of time available for testing. Usually, a power variation of a few per cent will give good results in a reasonably short time and will cause insignificant interference with normal operation. This would require a reactivity perturbation of a few cents. If it is necessary to use smaller reactivity perturbations, the testing time must be increased. For example, if the reactivity perturbation magnitude is decreased by a factor of 10, the testing time would have to be increased by a factor of 100 (see Section 4.9) to obtain the same noise error.

Because of the characteristics of digitizing equipment, the samples in a digitized data record are generally evenly spaced. Numerical integration procedures based on evenly spaced data are called Newton-Coates methods. A number of different Newton-Coates methods are available, and the effect of the choice of an integration method on the quality of the results should be examined.

The objective in calculating the DFT is to evaluate integrals of the form

where

or

t See Hunt (6).

We must make assumptions on the shape of y(t) between sampling points in formulating different numerical integration schemes. Each different inte­gration procedure is based on a different assumption on the shape of y(t) between samples. For the specific case of evaluating the DFT, we have complete knowledge of the behavior of one of the factors (sin cot or cos cot) in y(r). Thus we may develop our numerical integration procedure either by making an assumption on the form of y(t) between points or by making an assumption on x(t) between points.

Let us consider first the simplest numerical procedure, which involves a direct evaluation of Eq. (4.3.5). (We will use the complex form to simplify the discussion, but the reader will appreciate the equivalence with an analysis based on the trigonometric form.)

F'(jco) = (l/N) £ XP exp( ~jo)tp) (4.5.4)

P = 0

or

F’Ucj) = (l/N) £ xpe-^’

P = 0

This is equivalent to

F'(ja>) = X УР

p-0

y(t)

t

Fig. 4.8. Interpolation assumption in trapezoid rule.

where yp = (l/N)xpe~JapM. Now consider the numerical integration scheme based on the assumption that y(r) varies linearly between points (see Fig. 4.8). This is called the trapezoid rule. In this case,

f y(t) dt = {y(tp) + ±[y(tp+ j) — y(rp)]} Ar = |[y(tp+ j) + y(tp)] At (4.5.7)

J‘p

F(jco) = (l/N)^ xpe-^ (4.5.10) P= 0 is based on a trapezoid rule integration of the function x(t)e~jmt. We will call this the trapezoid DFT.

X(t)

f— 1

Fig. 4.9. Staircase interpolation of data

We now will consider other numerical DFT procedures that are based on assuming the form of x(t) between points. As a first choice, let us assume that x(t) is a staircase function (see Fig. 4.9). The DFT is given by

jco At

The second factor is identical with the trapezoid DFT result of Eq. (4.5.10). Thus we see that the staircase DFT is related to the trapezoid DFT by a simple factor. If the analyst has reason to believe that x(t) is nearly a stair­case function, then results from a trapezoid DFT may be transformed into the staircase DFT using Eq. (4.5.11).

Another approximation is that x(t) varies linearly between points. We will call this the ramp DFT. The reader should note that this is not the same as assuming that x(t) e~jat varies linearly between points as in the trapezoid DFT. It may be shown that an integration based on this assumption gives

This shows that the ramp DFT may be obtained by multiplying the trapezoid DFT by

(1 — e^’Xl — 2[1 — cos(w At)]

(со At)2 (со At)2 1 ‘

As with the staircase DFT, the ramp DFT may be obtained by multiplying the trapezoid DFT by a simple factor.

It should be noted that the correction factors will cancel if the ratio of two DFTs is calculated. Thus the corrections to the DFT are unnecessary in that case, but the individual DFTs may differ significantly from exact results.

The correction of the trapezoid DFT to give the staircase DFT and the ramp DFT may be demonstrated with two examples. Consider first a square wave. The staircase DFT should give exact results. (See Section 2.5 for an evaluation of the exact results.) Table 4.2 shows trapezoid DFT results and results obtained using Eqs. (4.5.11) and (4.5.9) to convert trapezoid DFT results into staircase DFT results and ramp DFT results for several different numbers of samples. The corrected staircase results agree perfectly with the exact theoretical values. A second example involves the triangular wave shown in Fig. 4.10. In this case, the ramp DFT should give exact results. Table 4.3 shows trapezoid DFT results and results obtained using Eq. (4.5.19). Again we find that the appropriate correction procedure gives the exact result.

This concluding chapter gives a brief summary of significant experiences with dynamics tests in nuclear reactors. An extensive bibliography is included so that the reader can find further information easily.

8.1. Pressurized Water Reactorsf

Several tests on pressurized-light-water reactors have been performed. Oscillator and PRBS tests were used at Saxton (20 MW) and PRBS tests were used at Yankee (600 MW) and Trino (825 MW). The experiences at Yankee and Trino are particularly significant because these tests were performed under conditions that are similar to conditions in new, larger PWRs. Rajagopal and Gallagher (5, 6) used normal system hardware to introduce 127-bit PRBS signals with bit durations of 0.5 and 6 sec. This work shows that the magnetic jack control rod drive mechanism (see Section 7.3) is capable of introducing large enough reactivity perturbations at a high enough rate to give frequency response results over the range of important frequencies for a PWR.

Rajagopal and Gallagher (5) also used their test results for system identifica­tion purposes. They approximated the feedback effects by two first-order lags. One was for the Doppler effect in the fuel and the other was for delayed

effects associated with the coolant and structure. They were able to evaluate the two gains and the two time constants in this simple model.

The autocorrelation function of a signal is defined as

(2.7.1)

where Tm is the integration time. The cross-correlation function of two signals is defined as

(2.7.2)

Clearly, if the signal at some time is strongly dependent on the same signal or another signal т seconds earlier, then the autocorrelation function or the cross-correlation function will be large for a delay t.

The autocorrelation function of a signal with period T or the cross-correla­tion function of two signals each with period T is a periodic function with period T.

It is important to develop a relation between the correlation functions and the Fourier coefficients for a signal. Consider first a periodic signal. It may be represented as

Then

or

Cі i(t) = (1/rj ехр[у(ш, + mk)t] dt

If Tm is an integer multiple of the period T, then this reduces to

Сц(т)= X Qc-texP(j’M

The Fourier transform of the autocorrelation function is

^{Сц(т)}Юк = (1/Г) Сц^ехрї-ушцтМт

That is, the Fourier transform of the autocorrelation function at frequency oik is the component of the power spectrum of the signal at frequency cok. The relation

= CkC_k (2.7.3)

is called Wiener’s theorem for periodic signals.

A similar relation may be developed for the cross-correlation function. If the Fourier series for x2(t) is

oo

x2U) = £ Dk exp(ja>kt)

fc = — oo

then

F{C 12(т)}Ик = DkC_k (2.7.4)

The quantity DkC^k is the component of the cross-power spectrum at cok.

Similar results may be derived for nonperiodic signals. In this case, the autocorrelation function and cross-correlation function are defined as

Л * m/2

Cn(r)= lim (1/TJ x^tjx^t + T)dt

Tm~* ZC J ~Tm/2

{•Tm/2

C12(t) = lim (l/TJ xi(t)x2(t + t)dt

The relations between the correlation functions and the power spectra are

f{Cn(r)) = x10’a))x1(-/ft)) (2.7.7)

^{Ci2W} = x^j^x^-jco) (2.7.8)

where XiO’ct)) is the Fourier transform of xt(t), and x2(jco) the Fourier transform of x2(t).

1.8. Convolution

The convolution theorem may be used for inversion of a product of Laplace transforms. The convolution theorem is

L~ 1{x1(s)x2(s)} = f хД-фc2(t — T)dr (2.8.1)

J о

where xt(t) is the inverse Laplace transform of x^s), and x2(t) the inverse Laplace transform of x2(s).

Example 2.8.1. Invert the following, using the convolution theorem:

1 / 1

s + 1) (s T 2

Xi(t) =

x2(t — r) = e~2t’~x)

f(t) = L lF(s) = Г e xe 2(‘ x> dx = e 1 — e 21 J о

The convolution integral has an important application in connection with system transfer functions. The definition of the transfer function G(s) is given by Eq. (2.3.1):

G(s) = SO(s)/Sl(s)

Then the Laplace transform of the output involves the Laplace transform of a product

SO(s) = G(s) SI(s)

Using the convolution theorem, we can specify the output time response: <50(t) = f h(t — t) <5/(t) dr (2.8.2)

SO(t) = Г /і(т) Sl(t — т) dr (2.8.3)

J о

where

/i(t) = L-1{G(s)} (2.8.4)

Thus, if h(t) is known, then the convolution theorem may be used to furnish the output for any input function. The function h(t) is called the impulse response of the system. It is a fundamental dynamic property of the system, and its determination is often the objective in a dynamics test.

The simplest space-dependent neutron kinetics model uses the mono­energetic diffusion equation

where D is the diffusion coefficient, ф the neutron flux, the macroscopic absorption cross section, Ly the macroscopic fission cross section, v the number of neutrons produced per fission, and v the average neutron velocity. Energy dependence of the neutron flux and the detection process can be handled using multiple energy groups.

In this book, we are not concerned with methods for calculating space — dependent frequency response functions. Rather, we are interested in the possible influence of spatial effects on experimental results.

Space-dependent effects are important in two significantly different cases. The first is a high-frequency effect that occurs because it takes a finite time for the flux to change from one steady-state distribution to another following a localized perturbation. This type of spatial effect is important only above 10 rad/sec. This is above the frequency range where feedback effects are important and where questions about the suitability of the system for normal operation are encountered.

The other spatial effect is a very-low-frequency phenomenon due to xenon-135. In this case, the flux is always essentially in a steady-state distribu­tion for a given core composition. The spatial effect occurs because the core composition changes with time because of changing xenon-135 concentration. The space dependence can occur because of the effect of the very large thermal absorption cross section of xenon-135 and because the xenon-135 is coupled to the neutron flux mainly through iodine-135. Iodine-135 is produced as a fission product, and it decays to xenon-135 with a 6.7-hr half-life.