**4. Quantitative Analysis of Results**

In this section, quantitative research is performed whose role is to define and determine important criteria parameters of the analyzed flow. It is difficult to find any favorable quantitative method in the literature on the subject of transient pipe flows. Here, we present a new methodology that results in two criteria parameters. The role of the final qualitative parameters is to determine the compliance of the simulated histories with respect to the experimental ones in a simplified mathematical way.

A MATLAB subprogram was written to search for maxima (peaks) and minimum values of pressure histories and their occurrence times (calculated from the beginning of the analyzed transient state). In a demonstrative way, Figure 6 illustrates the working idea of this proposed "collecting" subprogram. As can be seen, the pressure drops to the saturated vapor pressure were not taken into account, as the final results would be false. Additionally, when determining the time compliance, the time of the first pressure peak *t*<sup>1</sup> at the first amplitude was omitted, as it would also cause the final result to be distorted.

**Figure 6.** Selection of maximal and minimal pressures and their times of occurrence.

The pressure compliance parameter determining the compliance of the maximum and minimum simulated pressures is calculated by the following formula:

$$E\_p = \frac{\sum\_{i=1}^{k} \left| \frac{p\_{s,i} - p\_{c,i}}{p\_{c,i}} \right|}{k} \cdot 100\% \tag{57}$$

where *ps*,*i*—simulated maximal and minimal pressures and *pe*,*i*—experimentally predicted maximal and minimal pressures

The time compliance parameter that determines the time fit of subsequent simulated amplitudes was calculated using the following formula:

$$E\_l = \frac{\sum\_{i=2}^{k} \left| \frac{t\_{s,j} - t\_{c,j}}{t\_{c,j}} \right|}{k - 1} \cdot 100\% \tag{58}$$

where *ts*,*i*—times of occurrence of maximal and minimal simulated pressures and *te*,*i* experimentally observed times of maximal and minimal pressures (note that times *ts*,1 and *te*,1 representing the maximum at first amplitude are not taken into account, because in some cases, their fit could distort the whole analysis).

The degree of simulation compatibility increases with decreasing values of the above coefficients. In Table 3, the complete quantitative results of the *Ep* and *Et* parameters calculated for all comparative studies carried out in this work are summarized.


**Table 3.** Quantitative results.

The results of quantitative research indicate the following:

