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Since using the moving average method and the low-pass filter did not produce the expected analytical results from the cross-correlation and the second derivative, one of the ideas considered and applied to enhance the cross-correlation analysis was to use a technique known as’ the extended line’, as illustrated in Figure 7-17, which is applied after the signal filtering steps. The extended lines are based on parallel changes between two readings. For example, since the zoom of interest is primarily on the first wave, therefore the remaining data is trimmed and parallel lines are added to mimic the change in the behaviour between two points. The time delay is calculated and maintained since this is what should happen in reality and theoretically. The results of this technique were as predicted, improving the cross- correlation output on the delay and showing new peaks in some cases. Nevertheless, the peaks could not be correlated with any system features.

This approach was applied to various water hammer leak tests. For example, a leak at position 3 with steady state flow of 1.8 𝑙/𝑚𝑖𝑛𝑢𝑡𝑒 and the water hammer is

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triggered by shutting the manual downstream valve. The cross-correlation was done between P2 and P3, and also P3 and P4.

This approach depends on some alterations in the signal to improve the cross- correlation principle for extending the data with coherent signals with specific delay. The extension should make the correlation between the signals more robust, so the noise effect may be reduced and some outputs could be useful.

Referring to Figure 7-17, the period is time has been estimated for the two signals, the steps are as follows:

1. Divide the periodic of the time wave in half so that only the pressure surge is kept for further analysis.

2. Delete the remaining data from the trace and from the half point (point 1) draw diagonal line to 75% of the half period (point 2). The pressure of point 2 should be the initial steady pressure before the transient event.

3. The remaining 25% of the enhanced period should be the same as the start value, i.e. steady state pressure.

4. For the second pressure trace, the data should not be cut at the mid-point of the upsurge but further along the trace considering the time delay between the two signals.

5. The line for the second trace is drawn between the last point in the previous step (mid-point plus the time delay) and at the same slope as the previous line.

6. The second point can be obtained from the solving the line equation. The slope is now known and the line should be stopped at the initial value of signal two, so that the time can be defined.

7. After this time, the pressure value should be considered constant until the end of the total periodic time.

The aim of returning the values to the initial value for a short period of time (25% in the first signal and maybe less in the case of the second one) is to mimic the

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transient and improve the cross-correlation analysis. In summary, the enhanced lines sustain the time delay between the signals and the slope to maintain the signals’ coherency.

The above steps are summarised in Figure 7-17, which shows four graphs, each one having two arbitrary similar signals.

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The second derivative of the cross-correlation has been improved by using this approach. Despite the small offset shown near the middle because of the low-pass filter order, the offset between the signals is clearly evident. That is predictable, since the extended data retains the delay between the signals.

The following examples explain the approach. The two examples are transient events due to the pump start-up from zero to 1.8𝑙/𝑚𝑖𝑛𝑢𝑡𝑒 and closing the down stream valve for this steady state flow rate. First, the upstream event shown in Figure 7-18 shows the enhanced pressure readings for 𝑃2, 𝑃3 and 𝑃4. It is clear from Figure 7-18 that 𝑃3 has two different extended lines, because when comparing with 𝑃2, this will be the delayed signal, while with 𝑃4 the opposite is the case. As shown in Figure 7-18, sometimes maintaining the slope and returning to initial value is not possible, as in this figure. Sustaining the slope seems to be more reasonable and more important.

Figure 7-18: Enhanced measured pressure traces, for PT2, PT3, and PT4 for pump start-up from rest to 1.8 l/minute.

In Figure 7-19 and Figure 7-20 the second derivatives of cross-correlation for 𝑃2 − 𝑃3 and 𝑃3 − 𝑃4 are shown respectively, for the case of a leak at L3 and also no leak at L3.

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Figure 7-19: Second derivative for the cross-correlation between P2-P3 with enhanced extended lines.

Figure 7-20: Second derivative for the cross-correlation between P3-P4 L3 for the cases of no leak at L3 and leak at L3 water hammer due to pump star-up.

In the two figures, Figure 7-19and Figure 7-20 above, a noticeable improvement was achieved in the signal analysis to find the time delay between the two signals. However, the leak position it is not evident.

In conclusion with this enhanced line approach, whilst it has been applied successfully to the upstream water hammer to show the time delay between the signals, which we already know, the important feature, the position of the leak,

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cannot be found using this technique. However, in respect of the downstream transient, the time delay between the signals cannot be even identified in the second derivative of cross-correlation of the outputs.

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