*4.1. Comparison with Literature*

In this paragraph the results obtained from the presented analysis are compared with previous literature analyses. Different literature deterministic relationships for peak factor evaluation have been inspired by the following well known Babbitt's formula [7] deduced for domestic wastewater:

$$C\_{P,B} = \frac{5}{\left(\frac{N\_{\rm h}}{1000}\right)^{0.2}}\tag{13}$$

where *Nu* is the number of users, usually ranging between one thousand and one million in a population, as previously mentioned. The Babbitt's relationship was successively reformulated [28] as:

$$
\mu\_{\rm N\_u}(\Delta t) = K\_{\rm CP}(\Delta t) \times \frac{10}{N\_u \, ^{0.2}} \tag{14}
$$

Equation (14) is valid for 250 < *Nu* < 1250, and was originally obtained analysing data measured with a 1-min frequency (*KCP* = 1). *KCP* is a reduction coefficient that takes into account the effect of the time aggregation scale for time steps higher than 1 min. The results of the above relationships are compared with Equation (7), herein rewritten in terms of number of users *Nu* assuming that each meter serves 2.9 inhabitants on average, and considering the parameters corresponding to all the days of the week (Cluster 4 in Table 1):

$$
\mu\_{N\_u} = \frac{3.60}{N\_u^{0.67}} + 1.552\tag{15}
$$

Differently from Equation (15), Equations (13) and (14) do not exhibit any asymptote.

For *Nu* = 500 and *Nu* = 1000, Table 2 reports: (i) The values of the peak factor estimated by Equation (15); (ii) the values obtained by adopting Equation (13), and (iii) the values obtained by adopting Equation (14). In Equation (14), considering the experimental field data reported in [28], *KCP* is assumed to be equal to 0.65 for a sampling time step of 60 min. Table 2 highlights that the Babbitt's formula overestimates the peak factor [10], while the prediction obtained with the present analysis is comparable with the estimate of the formula proposed by [28]. In particular, the values obtained with Equation (14) are within the uncertainty range of Equation (7).

**Table 2.** Hourly peak factor values estimated with different relationships.


Forcing Equation (15) to assume a structure similar to Equation (14), it can be approximated by the following expression:

$$
\mu\_{\rm Nu} = \frac{4.3}{\mathcal{N}\_u^{0.18}} \tag{16}
$$

where the exponent for the number of users is very similar to the one in the empirical relationship in Equation (14) proposed by [28].

As previously mentioned, differently from the empirical literature relationships, the proposed Equation (7) for the evaluation of the hourly peak coefficient tends to an asymptotic value as the number of household increases. A similar result was also obtained by [16], who derived an estimation of the instantaneous peak factor using a probabilistic approach to describe the residential water use based on the Poisson Rectangular Pulse (PRP) model and adopting the Gumbel distribution for the extreme values. The asymptotic value can be assumed to be equal to the asymptotical hourly peak factor for a growing population [11]. Analyses performed on different towns in Italy showed that the asymptotic value ranges between 1.5 and 1.7 [11], similarly to the one deduced herein.

For a number of users varying between 3 and 3000, Equation (15) predicts a peak factor ranging between 3.3 and 1.55, which is the asymptotic value. Those values are also comparable with the range 1–5 reported in [26] considering the results of recent studies in different countries. The obtained values, smaller than the one provided by the empirical Babbitt's relationship, may be ascribed to a different kind of analysis and/or to a change in consumption behaviours compared to 30–50 years ago.
