Measurements by inverse point kinetics method

Experimental procedure

The inverse kinetic method was used for the estimation of control rod worth from control rod withdrawal (Δρ > 0) and control rod insertion experiments (Δρ < 0).

The experimental procedure for the control rod withdrawal measurements was already explained in the previous section (4.1.2). With respect to the control rod insertion measurements, they were all carried out adopting the procedure that follows. The reactor was brought to critical at powers between 10 and 15 W with both control rods and the start-up source withdrawn. The measured critical water level was typically of 952 mm at 20.0◦_{C. One control rod was completely inserted at full speed (< 1 s), and the power}

was allowed to decrease below 100 mW. The detection and acquisition system was the same as the one used for the control rod withdrawal measurements.

Since the control rod withdrawal and insertion experiments were performed having the reactor start-up source withdrawn, the inverse kinetics equation with no source (Eq. 4.4) was employed to estimate the reactivity worth as a function of the reactor power p(t) (or detectors’ count rate) and the reactor kinetic parameters βi, λi, Λ. A MatlabR script

was written to solve the inverse kinetics equation and to estimate the reactivity worth and associated uncertainty using random sampling techniques. The script processes the experimental data in the following manner:

1. The power signal p(t) given by the detectors’ count rate was down-sampled to 1 Hz (integration time of 1 s) to increase the number of counts per channel (hence reduce the statistical uncertainty) and to normalize all experimental data to a single sampling rate.

2. An algorithm allowed to select the time interval [ti, tf] after the rod insertion/with-

drawal where the power signal p(t) provides good counting statistics and the resulting reactivity function ρ(t) is reasonably invariant in time (see Fig. 4.8). The time interval was used to average the time-dependent reactivity ρ = _tf1_−ti tf

ti ρ(t)dt and it was computed as follows:

(a) For the rod withdrawal case, the time interval begins when the count rate exceeds 1× 103 cps (∼400 mW) and finishes 5 seconds before the end of the exponential growth.

(b) For the rod insertion case, the time interval begins 5 seconds after the rod drop and finishes when the count rate drops below 1× 103 cps (∼400 mW). 3. Independent probability distributions of S samples were randomly generated for the

input parameters. A Poisson distribution was used for the power p(t) and normal distributions were used for the kinetic parameters βi, λi, Λ.

4. The probability density functions (PDF) for each input quantity were randomly propagated through the inverse kinetics equation to obtain a PDF for the reactivity. The reactivity output PDF is contained within a matrix of size T × S, where T is the length of the power signal sampled at 1 Hz during the time interval [ti, tf] and

S is the number of random samples.

5. The expected reactivity value was computed as the arithmetic mean of all samples from the output probability distribution E(ρ) = ˜ρ. The associated standard uncertainty was computed as the standard deviation of the output PDF u(ρ) = σ(ρ). Figure 4.8 shows an example of the power response following a rod insertion and the calculated reactivity by inverse kinetics. The figure also shows the time interval [ti, tf]

where the reactivity was averaged to obtain an estimate of the reactivity worth. The uncertainty propagation by random sampling yields a probability density function for the reactivity similar to that one showed for the asymptotic period method in Figure 4.7.

Results

Measurements for control rod insertion were repeated three times for the South East (SE) control rod, and six times for the North West (NW). Table 4.2 shows the full list measurements using the inverse kinetics method. Uncertainties were calculated in an equivalent way than for the asymptotic period measurements. They are expressed with a coverage factor of k = 1, providing a confidence level of ∼ 68%.

100 200 300 400 500 600 700 time [s] 101 102 103 104 105

Detector count rate [cps]

-0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 Reactivity [$] reactor power (detector counts) reactivity (inverse kinetics)

Figure 4.8 – Example rod insertion measurement and reactivity estimation by inverse kinetics.

Table 4.2 – Reactivity worth by IK method - Rod insertion experiments Run # Control rod Detector Reactivity worth∗ [$]

12a NW FC North -0.233 ± 0.023 (0.005) 12b NW FC South -0.228 ± 0.031 (0.005) 13b SE FC South -0.235 ± 0.014 (0.003) 14a NW FC North -0.232 ± 0.013 (0.005) 14b NW FC South -0.233 ± 0.015 (0.005) 15a SE FC North -0.241 ± 0.015 (0.003) 15b SE FC South -0.239 ± 0.016 (0.003) 16a NW FC North -0.240 ± 0.013 (0.005) 16b NW FC South -0.240 ± 0.014 (0.005)

∗_{Result expressed as: ρ ± uT (σexp).}

It is worthwhile noting that the experimental conditions for rod insertion experiments were not always optimal because, in some cases, the reactor power before the rod drop was not high enough to provide good counting statistics. The experimental run #12 (see Table 4.2) exemplifies the case where the initial power before the rod insertion was low (in the order of 1 W). Due to the low power, the detectors’ signal –that is proportional to the power p(t)– carried a large statistical noise that was propagated towards the final estimation of reactivity. The whole set of rod insertion experiments was executed at lower power ranges (15 to 0.4 W) than the withdrawal ones (0.4 to 30 W), and therefore all reactivity values from rod insertion measurements carried larger uncertainties than their withdrawal counterparts.

also used to estimate the reactivity worth by inverse kinetics. These results are listed in Table 4.3 and summarized in Section 4.1.5.

Table 4.3 – Reactivity worth by IK method - Rod withdrawal experiments Run # Control rod Detector Reactivity worth∗ [$]

1a SE FC North 0.232± 0.007 (0.001) 1b SE FC South 0.233± 0.008 (0.001) 2b NW FC South 0.233± 0.008 (0.002) 3a SE FC North 0.229± 0.007 (0.001) 3b SE FC South 0.229± 0.008 (0.001) 4a NW FC North 0.236± 0.007 (0.002) 4b NW FC South 0.236± 0.009 (0.002) 5a SE FC North 0.230± 0.007 (0.001) 5b SE FC South 0.231± 0.008 (0.001) 6a SE FC North 0.231± 0.008 (0.001) 6b SE FC South 0.231± 0.007 (0.001) 7a SE FC North 0.231± 0.007 (0.001) 7b SE FC South 0.230± 0.007 (0.001) 8a SE FC North 0.231± 0.008 (0.001) 8b SE FC South 0.231± 0.008 (0.001) 9a SE FC North 0.232± 0.007 (0.001) 9b SE FC South 0.232± 0.008 (0.001) 10a SE FC North 0.232± 0.007 (0.001) 10b SE FC South 0.232± 0.008 (0.001) 11a SE FC North 0.231± 0.007 (0.001) 11b SE FC South 0.231± 0.008 (0.001)

∗_{Result expressed as: ρ ± uT (σexp).}