Introduction

The water treatment plant (WTP) of the city of Torino (NW Italy) treats about 40 · 106 m3/year of raw water from Po river. While a half of the inflow is sent directly to the WTP, the other half undergoes a lagooning process in a 15-ha artificial basin. Lagooning is a process that employs natural or artificial basins to perform a pre-treatment on waters to be destined to WTPs or wastewater treatment plants (WWTPs). The latter use is far more widespread than the first, especially in rural regions or tropical countries, thanks to its low costs and ability to handle fluctuating hydraulic and organic loads (Mara 2013). During the permanence in lagooning basins, wastewaters receive treatment through a series of physical, biological, and biochemical processes. Most of the treatments occur naturally, but some systems are also designed to use aeration devices in order to increase the amount of oxygen in the wastewater and accelerate biological processes. On the other hand, in the field of water treatment for human consumption, lagooning is used with three main purposes:

  • homogenization of the physical and chemical characteristics of the water flow that has to be sent to the WTP with the aim of keeping a constant dosage of chemicals;

  • water self-purification with the reduction of chemical and biological pollutant load and turbidity (due to the suspended solid content), subsequent improvement of the raw water quality, and decrease in the chemical consumption;

  • emergency feed of the WTP in case of accidental contamination of the river or low water.

With reference to the first of the aspects listed above, the efficiency of a basin to smooth and homogenize water flows with anomalous physical and chemical characteristics mainly depends on the internal hydrodynamic behavior of the basin. The hydrodynamic behavior of the basin, in turn, is influenced by some factors such as shape and depth of the basin, characteristics of inlet and outlet, climatology, and direction of dominant winds in the zone where the basin is situated (Torres et al. 1999).

The internal hydrodynamics control the flow patterns in the basin thus influencing the actual residence time (RT) and the efficiency of water mixing. The value of the actual RT depends on the presence of possible anomalies in the flow, such as dead zones or short circuits (Levenspiel 1999). On the other hand, by making reference to a 1-D advection–dispersion hydrodynamic model, the aspect which concerns the water mixing can be described as plug flow or completely mixed or an intermediate situation, known as “dispersed” plug flow system, and can be quantified by a single dimensionless parameter, known as dispersion number (Arceivala 1981; Polprasert and Bhattarai 1985; Nameche and Vasel 1998; Makinia and Wells 2005). The dispersion number (or mixing parameter or inverse Peclet number, Pe, dimensionless) is defined as the ratio between D x and u · L, where D x is the dispersion coefficient in the axial direction (m2/day), u is the average longitudinal velocity (m/day), and L is the length of the basin in the axial direction (m). If the dispersion number tends toward zero, with values of less than 0.05–0.2, the reactor approximates to the ideal plug flow. On the other hand, when the dispersion number is greater than 0.5–4, complete mixing may be assumed.

Several authors have simulated the flow patterns into a basin with computational fluid dynamics (CFD) software with the aim of obtaining the two parameters which describe the hydraulic characteristics of the basin, that is the actual RT and the dispersion number (Wei et al. 2013; Barrio et al. 2015; Singh et al. 2015). According to another approach, the two parameters can be derived using tracer tests that are a good methodology for the characterization of the hydraulic properties of reactors, although often quite lengthy and costly, especially when applied to large systems (Williams and Nelson 2011; Bodin et al. 2012; Liang et al. 2013; Bloem et al. 2014; Wang et al. 2014). This is the reason why, despite the major influence of pond hydraulic characteristics on their efficiency in lagooning processes, very few tracer studies have been carried out on real-scale facilities and the literature pertaining to studies on the hydraulics of basins is sparse and fragmentary.

In most of the existing experiences that refer to tracer tests, the authors have attempted to derive the parameters which describe the internal hydrodynamics of the basins from the E-curves (or residence time distribution (RTD) curve, i.e., the output from a stimulus–response tracer test) (Burrows et al. 1999; Makinia and Wells 2005; Teixeira Costa and Siqueira do Nascimento 2008). Only few studies (Torres et al. 1999; Delatolla and Babarutsi 2005; Broughton and Shilton 2012) have monitored the tracer concentration in the basin with the aim of matching the trend of the tracer record at the outlet channel and the distribution of the tracer throughout the basin at different times during the experimentation.

In this paper, the internal hydrodynamics of a basin with a surface of about 15 ha and a capacity of 1.8 · 106 m3, used as a lagooning pre-treatment facility in the WTP of the city of Torino, was studied by combining the results of a stimulus–response tracer test with the monitoring of the tracer (fluoride) concentration in the lagoon on different times.

The efficiency of the lagooning process in smoothing peaks of the concentration of contaminants and homogenizing physical and chemical characteristics of the raw water that has to be sent to the WTP was assessed by:

  1. 1.

    deriving the actual RT from the first momentum of the RTD curve and comparing it with the theoretical RT, in order to detect the presence of flow anomalies (short circuits, dead zones);

  2. 2.

    assessing the congruence between the tracer concentration values recorded at the outlet channel of the basin and those found throughout the basin on the monitoring dates by employing a mass balance and a continuous stirred tank reactor (CSTR) model;

  3. 3.

    deriving the dispersion number from the second momentum of the RTD curve and comparing it with the values calculated from various theoretical and empirical models which have been developed by several authors for ponds and aerated lagoons;

  4. 4.

    comparing the distribution of the tracer at the outlet channel and throughout the basin with the values provided by the 1-D and 2-D advection–dispersion models.

Materials and methods

Site description and sampling campaign

The study was carried out on a lagoon of 15.2 ha, 1,787,950 m3, located in La Loggia (44°57′31″ N, 7°40′06″ E), near the town of Torino, north-west of Italy. The average daily flow throughout the basin during the period of the study was 1,150 L/s, with a nominal RT of 18 days. The length of the basin in the axial direction (L) was 700 m, the mean width in the transverse direction (W) approximately 220 m, and the mean depth (Z) 11.8 m. The tracer test was carried out in the Fall season (October) when, in the basin, no thermal stratification was observed.

For the tracer study, an amount of NaF equal to 180 kg was employed. The salt was diluted with about 50,000 m3 of the lagoon water and introduced into the basin in 12 h, in order to simulate a pulse injection. The amount of the tracer added was fixed in order not to exceed the threshold concentration of 1.5 mg/L (maximum concentration value allowed in drinking water, Italian D.Lgs. 31/01), under the hypothesis that the lagoon was not mixed at all (plug flow modality).

Fluoride concentration values at the outlet channel of the basin were recorded using a fluoride selective probe (ISE25F, Radiometer Analytical), every 45 min, until day 29 from the introduction of the tracer, when the observed concentration values were lower than the probe detection limit. Samples were also collected at 15 points (see Fig. 1) over 3 depth values, for a total of 45 sampling points throughout the lagoon, 1, 7, and 14 days after the introduction of the tracer. The determination of the fluoride concentration in the samples collected throughout the basin was performed using a Dionex DX 320 ion chromatograph.

Fig. 1
figure 1

Schematic representation of the basin and fifteen sampling points

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RTD curve development and actual RT calculation

The registration of the fluoride concentration versus time at the outlet channel of the basin produces the RTD curve or E-curve (Levenspiel 1999), which is the record of the length of time that each fluid element takes to pass through the system. RTD curves are usually normalized in order to permit the comparison of the results from different experimentation campaigns (Teixeira Costa and Siqueira do Nascimento 2008). The normalization is done by dividing the measured concentrations (C) by the initial average concentration (C 0), which is the ratio between the mass of the tracer that was injected into the flow and the effective volume of the basin, and the time (t) by the theoretical retention time (τ), which is the ratio between the effective volume (V) and the flow rate (Q). The fluoride concentration values recorded at the outlet channel of the basin were averaged on a daily base, in order to obtain a single concentration value for each day.

According to Levenspiel (1999), the actual mean RT \( \left(\overline{t}\right) \) of the fluid in the basin could be calculated by determining the value of the first moment of the RTD curve, as in Eq. (1):

$$ \overline{t} = \frac{{\displaystyle {\int}_0^{\infty }t\cdotp C\ dt}}{{\displaystyle {\int}_0^{\infty }}C\ dt} $$
(1)

Comparison between the amount of the tracer found in the lagoon and that predicted by a mass balance

The congruence between the tracer concentration values observed at the outlet channel of the basin and those found throughout the basin on the three monitoring dates was assessed by employing a mass balance. The amount of fluoride found in the basin during the sampling campaign, 1, 7 and 14 days after the introduction of the tracer, was calculated by multiplying each of the 45 concentration values by the portion of the basin volume of competence. The obtained values were compared with the amounts of fluoride resulting from the mass balance equation, as in Eq. (2):

$$ {M}_{\mathrm{i}} = {M}_{\mathrm{i}-1} - Q\cdotp {C}_{\mathrm{i}-1}\cdotp \varDelta t $$
(2)

where:

M i :

is the mass of fluoride in the basin on the i-day (with i = 1 ÷ 29)

M i-1 :

is the mass of fluoride in the basin on the (i-1)-day

Q :

is the inflow and outflow rate, equal to 1150 L/s

C i-1 :

is the concentration of fluoride at the outlet channel of the basin on the (i-1)day

Δt :

is the discretization time, equal to 1 day.

The balance described in Eq. (2) may be applied under the hypothesis that the concentration at the outlet channel of the basin, on the i-day, is constant for the entire length of the day (that is, the concentration recorded by the fluoride selective probe every 45 min is daily averaged).

Application of the CSTR Model

The data obtained as in the “Comparison between the amount of the tracer found in the lagoon and that predicted by a mass balance” section were compared with the data gathered from the CSTR model. The application of the CSTR model considers that the mass of fluoride introduced in the basin (68,400 g) was perfectly mixed with the active volume to obtain the corrected initial average concentration (C0′ = 54.3 μg/l), with an identical concentration value in every point of the basin and, consequently, at the outlet channel of the basin. The discretization time employed was 1 day.

Determination of the axial dispersion coefficient from the RTD curve and comparison with empirical formulae

In a 1-D advection–dispersion model, the mixing conditions of the basin may be described by the dispersion number that is in turn dependent on the axial dispersion coefficient D x. According to several authors (Nameche and Vasel 1998; Burrows et al. 1999; Levenspiel 1999; Baléo et al. 2001; Makinia and Wells 2005), the 1-D advection–dispersion model is based on the ideal plug flow model and the deviations from it caused by back mixing or random fluctuations. By applying Fick’s law in the longitudinal (x) direction only and assuming steady state conditions, Eq. (3) describes the concentration for a general point in a tank at time t,

$$ {D}_x\frac{\partial^2{C}_t}{\partial {x}^2}=u\frac{\partial {C}_t}{\partial x}+\frac{\partial {C}_t}{\partial t} $$
(3)

where D x is the axial dispersion coefficient, C t the concentration of tracer at time t, and u the average longitudinal velocity.

According to Levenspiel (1999), the dispersion number is directly related to the second momentum, that is the variance, of the RTD curve (σ t 2). The RTD curve obtained in a tracer study with an impulse signal can be used to estimate the D x value by means of the relationship between σ t 2 and D x using Laplace transforms for a closed system (D x (∂C/∂x) = 0 at inlet and outlet) and constant D x value through the basin (Makinia and Wells 2005):

$$ {\sigma}_t^2=2\frac{D_x}{uL}-2\left(\frac{D_x}{uL}\right)\left[1- \exp \left(-\frac{uL}{D_x}\right)\right] $$
(4)

The variance, σ t 2, for any experimental response curve can be calculated from a dimensionless plot of concentration and time, as in Eq. (5):

$$ {\sigma}_t^2=\frac{{\displaystyle {\int}_0^{\infty }}{\left(\frac{t}{\overline{t}}-1\right)}^2Cdt}{{\displaystyle {\int}_0^{\infty }}Cdt} $$
(5)

where \( \overline{t} \) is the actual mean RT for the C vs. t distribution. Combining and rearranging Eqs. (4) and (5), a value of the dispersion coefficient D x can be calculated from the field data of C and t.

The value of the axial dispersion coefficient obtained as in Eq. (4) was then compared with values of the same parameter calculated using the literature models developed by Arceivala (1981), Polprasert and Bhattarai (1985), and Nameche and Vasel (1998).

Arceivala (1981) proposed four equations, depending on the geometric shape of the basin and the presence of baffles, based on studies of correlation between various geometric and flow parameters and the axial dispersion coefficient. In this work, the formula for basins with the mean width in the transverse direction (W) of more than 30 m, without baffles, was considered.

$$ {D}_x\left({\mathrm{m}}^2/\mathrm{h}\right) = 33 \cdot p\ W\ \left(\mathrm{m}\right) $$
(6)

Polprasert and Bhattarai’s formula (1985) relates the dispersion number to the theoretical RT (τ, days), fluid viscosity (ν, m2/s) and pond geometry, as in Eq. (7):

$$ \frac{1}{Pe}=\frac{0.184\cdot {\left[\tau \cdot v\left(W+2Z\right)\right]}^{0.489}{W}^{1.511}}{(LZ)^{1.489}} $$
(7)

where L (m) is the length of the basin in the axial direction, W (m) the mean width in the transverse direction, and Z (m) the mean depth.

Toward the end of the Nineties, after an examination of the existing dispersion number prediction formulae, Nameche and Vasel (1998) proposed three new empirical equations based on tracer studies performed at thirty full scale facilities, relating the Pe number to dimensional ratios and, if present, the input aerator power. In this work, among the three formulas, that obtained for “all basin” and reported as in Eq. (8) was employed to calculate the dispersion number

$$ \mathrm{P}\mathrm{e} = 0.35 \cdot p\ L/W + 0.012 \cdot p\ L/Z $$
(8)

where, as in Polprasert and Bhattarai’s formula, L (m) is the length of the basin in the axial direction, W (m) the mean width in the transverse direction, and Z (m) the mean depth.

1-D modeling

The D x value obtained from the second momentum of the RTD curve was employed to calculate the values of fluoride concentration over time at the outlet channel of the basin according to a 1-D advection–dispersion model. The calculation was done by referring to the analytical solution of the 1-D model (see Eq. 3), that is

$$ c\left(x,t\right)=\frac{M}{\sqrt{4\pi {D}_xt}} \exp \left[-\frac{{\left(x-ut\right)}^2}{4{D}_xt}\right] $$
(9)

In Eq. (9) M was the tracer mass per unit of transversal surface and u is the actual water velocity in the lagoon along the longitudinal direction. Both values were derived from the actual RT.

Curves at the outlet channel of the basin were also calculated for D x  = 10,000; 20,000; 30,000; 100,000 m2/day and compared with the record of the experimental data.

2-D modeling

A 2-D model was employed to gather the dispersion coefficients in the x and y directions.

$$ {D}_x\frac{\partial^2{C}_t}{\partial {x}^2}+{D}_y\frac{\partial^2{C}_t}{\partial {y}^2}=u\frac{\partial {C}_t}{\partial x}+\frac{\partial {C}_t}{\partial t} $$
(10)

The dispersion coefficients along the x and y directions were obtained by comparing the outline of the tracer concentration over time at the outlet channel of the basin with the best fit calculated from the 2-D model. Under the hypotheses of a linear vertical source and a pulse injection, the 2-D advection–dispersion model (Eq. 10) can be solved by Eq. (11):

$$ c\left(x,y,t\right)=\frac{m}{4\pi bt\sqrt{D_x{D}_y}} \exp \left[-\frac{{\left(x-ut\right)}^2}{4{D}_xt}-\frac{y^2}{4{D}_yt}\right] $$
(11)

where:

  • m, amount of fluoride introduced in the lagoon, 68,400 g;

  • b, vertical extent of the fluoride source, i.e., lagoon depth, 11.8 m (calculated as the ratio between the volume and the surface of the lagoon);

  • u, actual water velocity in the lagoon, derived from the actual residence time, 55.2 m/day;

  • D x and D y , axial and transverse dispersion coefficient, unknown (m2/day).

The two calculated dispersion coefficients were employed to make a comparison between the real distribution of the tracer concentration in the basin and the distribution predicted by the analytical solution of the 2-D advection–dispersion model. In order to calculate the tracer concentration in the fifteen sampling points, the coordinates of each point in the (N, E) reference system were translated in a (X, Y) system, employing the set of equations as in (12a) and (12b):

$$ \mathrm{X} = \mathrm{E}\ \left(- \sin \upalpha \right) + \mathrm{N}\ \cos \upalpha $$
(12a)
$$ \mathrm{Y} = \mathrm{E}\ \cos \upalpha + \mathrm{N}\ \sin \upalpha $$
(12b)

the center of which corresponds to the inlet of the basin and the x-axis, rotated by 25° anticlockwise (α) with respect to the (N, E) reference system, coincides with the axial direction of the basin.

Results and discussion

RTD curve development and actual RT calculation

The initial average concentration, C 0, as defined in the “RTD curve development and actual RT calculation” section, was equal to 38.3 μg/L. This value was obtained by dividing the mass of fluoride introduced into the lagoon, 68,400 g, by the volume of the basin, 1,787,950 m3. As recalled in the “Site description and sampling campaign” section, the nominal RT (or theoretical retention time, τ) of the basin was 18 days. The normalized RTD curve is shown in Fig. 2. The actual mean RT \( \left(\overline{t}\right) \) was calculated from the value of the first moment of the RTD curve, as in Eq. (1), and it was equal to approximately 12.7 days.

Fig. 2
figure 2

Normalized hydraulic residence time distribution (RTD)

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According to the findings from previous tracer tests carried out on wastewater treatment basins (Delatolla and Babarutsi 2005; Teixeira Costa and Siqueira do Nascimento 2008; Broughton and Shilton 2012), the value of the calculated actual mean RT is consistent when almost 100 % of the tracer is recorded at the exit of the basin. In fact, according to Broughton and Shilton (2012), a common mistake in tracer studies is that they are terminated too early and the missing tracer leads to misreporting of the mean RT and, as a consequence, of the dead space in the basin. Even though, according to Delatolla and Babarutsi (2005), there is currently no consensus on the correct method of developing the tail portion of a RTD curve, Levenspiel (1999) illustrated that in systems with non-ideal flow patterns such as dead zones, the tracer concentration at the outlet often begins to decrease exponentially after a certain amount of time elapses.

For the calculation of \( \overline{t} \) performed in this work, it had to be taken into account that the mass of fluoride ion at the end of the monitoring campaign was still 7,550 g, slightly less than 12 % of the amount introduced. This value was returned by the mass balance on the fluoride ion calculated as in Eq. (2). The results of the mass balance calculation are shown in Fig. 3 and discussed in detail afterwards.

Fig. 3
figure 3

Comparison between the real amount of fluoride ion found in the basin (at days 1, 7, and 14 after the tracer introduction) and the amounts calculated from a mass balance and by applying the CSTR model (u = 38.9 m/day and u′ = 55.2 m/day)

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For this reason, it was necessary to develop the tail portion of the RTD curve from day 30 until the moment in which the residual amount of fluoride was less than 5 % of the mass introduced. The tail portion of the RTD curve was developed by interpolating the experimental data having a descending trend (i.e., from day 3 up to day 29 from introduction of the tracer) with a first order rate law (see Fig. 4). Among different rate law equations (linear, logarithmic, polynomial), the first order rate law returned the best fitting of the experimental data and was in agreement with Levenspiel’s statement reported above (Levenspiel 1999). The equation of the best-fit exponential curve was

Fig. 4
figure 4

Concentration of fluoride at the outlet channel of the basin and best fit curve

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$$ C = 0.0563\cdotp {\mathrm{e}}^{-0.076\cdotp t}\kern0.5em \left({R}^2 = 0.9638\right) $$

The tail portion of the curve was then developed from day 29 up to day 44, when the residual mass of fluoride in the lagoon was about 3,000 g, a value of less than 5 % of the amount introduced.

The actual mean RT at the outlet channel of the basin, where closed conditions are respected, can be compared with the theoretical RT, τ. This comparison gives some information about the flow pattern such as the magnitude of the dead volume or the short circuiting flow rate. Thus, a value of \( \overline{t}/\tau \) <1 indicates the existence of a dead volume, while a short circuiting flow rate exists if the same ratio is greater than unit. In the case presented in this work the ratio \( \overline{t}/\tau \) was 0.70. According to Delatolla and Babarutsi (2005), the basin active volume, Va, is calculated using the equation Va = Qa · \( \overline{t} \), where Qa is the active flow, usually equal to the theoretical flow, Q. Therefore, the basin active volume was approximately 1,270,000 m3, 70 % of the theoretical volume.

Mass balance development and CSTR modeling

The results of the lagoon monitoring at days 1, 7, and 14 after the introduction of the tracer, processed by means of the Surfer software (Golden Software Inc.), are shown in Fig. 5. The distribution of the fluoride concentration in the basin during the monitoring period was such that a good mixing in the whole basin volume may be assumed.

Fig. 5
figure 5

Results of the lagoon monitoring at days 1, 7, and 14 (respectively, a, b and c), after the tracer introduction. Tracer concentration values at days 1, 7, and 14 (respectively, d, e and f), after the tracer introduction, calculated using the analytical solution of the 2-D advection dispersion model (u = 55.2 m/day; D x  = 27,000 m2/day; D y  = 110 m2/day). Data processing was carried out using Surfer software

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The amount of fluoride detected in the basin during the sampling campaign, 1, 7, and 14 days after the tracer introduction, was then compared with the amount calculated using the mass balance developed as in Eq. (2). As shown in Fig. 3, there was a very good agreement between the values predicted by the mass balance and those found in the lagoon. In particular, with reference to the three couples of values found at days 1, 7, and 14 after the introduction of the tracer, it could be seen that at day 1, the mass balance overestimated the real amount by about 3.5 %. On the other hand, the values calculated at days 7 and 14 were lower than the real ones by about 16 % and 10 %, respectively (see Fig. 3).

Figure 3 also shows the trend of the tracer mass inside the basin predicted by the CSTR model. The calculation was done considering both the geometrical volume of the basin (with a corresponding velocity of 38.9 m/day) and the active volume (with a corresponding velocity of 55.2 m/day). The trends of the curves reported in Fig. 3 highlight a good correspondence between the masses of fluoride calculated from the mass balance described as in Eq. (2) and the masses of fluoride predicted by the CSTR model applied to the active volume. These observations lead to the conclusion that dissolved contaminants (such as fluoride) that may be brought to the WTP by the river could be efficiently diluted by the lagooning basin before entering the WTP.

Evaluation of the axial dispersion coefficient from the RTD curve and comparison with empirical formulae

To complete the hydraulic characterization of the basin, the axial dispersion coefficient was evaluated. With reference to the determination of the dispersion number and the axial dispersion coefficient, the variance of the RTD calculated as in Eq. (5) resulted to be equal to 1.08. Consequently, the dispersion number was 1.22 and the axial dispersion coefficient was 47,260 m2/day, considering the actual longitudinal velocity equal to 55.2 m/day and the length of the basin in the axial direction of 700 m.

According to Burrows et al. (1999), a basin which approximates to a single CSTR will have a larger dispersion number (greater than 0.2), indicating a high degree of longitudinal mixing. The obtained value of the dispersion number is in line with both the trend of Fig. 3 and the distribution of the fluoride concentration in the basin shown in Fig. 5, thus suggesting that the behavior of the basin could be reliably described by a CSTR model.

Table 1 reports the values of the axial dispersion coefficient obtained from the outcomes of the tracer test and by applying the formulae developed by Arceivala (1981), Polprasert and Bhattarai (1985), and Nameche and Vasel (1998). As shown in Table 1, in the case presented in this work, Nameche and Vasel’s equation showed better results over the previous empirical models (Arceivala; Polprasert and Bhattarai). The weakness of these two formulae was demonstrated by Marecos do Monte and Mara (1987), who pointed out that there was not a good agreement between the dispersion number gathered from tracer data and the models presented by Arceivala (1981) and Polprasert and Bhattarai (1985), thus showing the limitations of the empirical techniques for the calculation of the dispersion number. However, the set of equations by Nameche and Vasel (1998) also suffered from the fact that they are purely statistical and there is no physical basis for the relationship.

Table 1 Axial dispersion coefficient values (from the tracer test and literature formulae)
Full size table

1-D modeling

The D x value obtained from the second momentum of the RTD curve (47,260 m2/day) was employed to evaluate the trend of fluoride concentration over time at the outlet channel of the basin under a 1-D advection–dispersion model (see Eq. 3). Curves of fluoride concentration at the outlet channel of the basin were calculated also for D x  = 10,000; 20,000; 30,000; and 100,000 m2/day.

Making reference to Eq. (9),

$$ c\left(x,t\right)=\frac{M}{\sqrt{4\pi {D}_xt}} \exp \left[-\frac{{\left(x-ut\right)}^2}{4{D}_xt}\right] $$

M was the tracer mass per unit of transversal surface, 38.0 g/m2, and u the actual water velocity in the lagoon along the longitudinal direction, 55.2 m/day. Both these values were obtained by taking into account that only 70 % of the basin volume may be considered as active.

Figure 6 reports the outline of the tracer concentration over time at the outlet channel of the basin, calculated, as in Eq. (9), for D x  = 10,000; 20,000; 30,000; 47,260; and 100,000 m2/day. For the aforementioned values of the axial dispersion coefficient, the dispersion numbers, calculated as the ratio between D x and u · L (38,650 m2/day), resulted respectively equal to 0.259; 0.517; 0.776; 1.22; and 2.59. According to the 1-D advection–dispersion model, the tracer immediately spreads along a surface of about 1800 m2, to reach a uniform concentration value, and it begins to move from there in the axial direction, being subjected to the advection–dispersion phenomena. According to what is reported in the “Introduction” section, for D x values of less than 10,000 m2/day the reactor approximates to the ideal plug flow. All the other curves (for D x from 20,000 to 100,000 m2/day) referred to well mixed scenarios.

Fig. 6
figure 6

Outline of the tracer concentration in time at the outlet channel of the basin: experimental data and curves calculated from the analytical solution of the 1-D advection-dispersion model for D x  = 10,000; 20,000; 30,000; 47,260; and 100,000 m2/day

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The set of the calculated curves was compared with the outline of the experimental tracer concentrations recorded at the outlet channel of the basin. As in Fig. 6, there was not a good correspondence between the curves obtained by applying the 1-D advection–dispersion model and the outline of the fluoride concentration over time at the outlet channel of the basin. The marked difference between the experimental curve and the set of curves obtained from the model may be due to the inappropriateness of the model employed, which was a 1-D. In fact, the conditions required for the appropriate application on the afore-mentioned model are that (1) the longitudinal dispersion must be independent of position, (2) the transverse dispersion must be very high with respect to the longitudinal dispersion, and (3) there should be no flow variations along the flow path. In the case of the basin where the tracer test was carried out, the conditions number (1) and (3) were probably fulfilled but because of the width extent (about 220 m in the transverse direction compared with 700 m in the axial direction) and the characteristics of the basin inflow, condition (2) was far from being complied.

2-D modeling

Due to the failure of the 1-D advection–dispersion model, a 2-D model was employed to gather the axial (x direction) and transverse (y direction) dispersion coefficients. The dispersion coefficients along the x and y directions were obtained by comparing the outline of the tracer concentration over time at the outlet channel of the basin, with the best fit calculated from the 2-D model (see Fig. 7).

Fig. 7
figure 7

Outline of the tracer concentration in time at the outlet channel of the basin: experimental data and curves calculated from the analytical solution of the 2-D advection–dispersion model for D x  = 27,000 m2/day and D y  = 110 m2/day

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The best D x and D y coefficients were found at 27,000 m2/day and 110 m2/day, respectively. The two dispersion coefficients were employed to make a comparison between the real distribution of the tracer concentration in the basin (as shown in Fig. 5) and the distribution predicted by the analytical solution of the 2-D advection–dispersion model.

The distribution of the tracer concentration 1, 7, and 14 days after its introduction into the basin, predicted by the 2-D model, is shown in Fig. 5d–f and detailed for each sampling point in Table 2. Although the correspondence between the experimental data and those predicted by the 2-D advection–dispersion model at the outlet channel of the basin was good (see Fig. 7), this was not true for the distribution of the tracer in the basin. That might be due to two main reasons:

Table 2 Values of the tracer concentration at the 15 sampling points, 1, 7, and 14 days after its introduction in the basin from the measuring campaign and predicted by the 2-D model
Full size table

1. If short times from the tracer injection are considered (i.e., one day), it is possible that the value of D y returned by the analytical solution of the 2-D model was too small to be capable of describing the quick mixing of the tracer effectively observed along the y direction. The D y value was in fact of about two orders of magnitude lower than the dispersion coefficient in the x direction;

2. On the other hand, for longer times from the tracer injection (i.e., 7 and 14 days), it has to be taken into account that the 2-D model employed did not consider the limitation of the domain in the y direction, thus providing concentration values lower than the real ones.

In order to simulate the distribution of the tracer in the basin, a 3D model should have been employed. Such a model could take into account the effect of the basin depth, responsible for the delay in the tracer mixing along the z-axis. On the other hand, the domain limitation along the y direction could be modeled by using appropriate virtual tracer sources.

Conclusions

This study was aimed at describing the hydrodynamic behavior of a basin employed as a lagooning pre-treatment facility by the WTP of the city of Torino (NW Italy). The internal hydrodynamic behavior of the basin strongly affects the efficiency of the lagooning process in smoothing peaks of concentration in order to send water flows with near constant characteristics to the WTP.

The concentrations of fluoride from the tracer test recorded at the outlet channel and inside the basin demonstrated that the system was efficiently mixed without relevant flow anomalies such as dead zones or short circuits. In fact:

  • the first detection of the tracer at the outlet channel of the basin occurred after very few hours from the beginning of the test;

  • the mean RT, in the presence of the flow rate employed for the test, was about 13 days compared with a theoretical hydraulic RT of 18 days;

  • both the obtained value of the dispersion number (1.22) and the distribution of the fluoride concentration in the basin during the monitoring period suggested that the behavior of the lagooning basin could be reliably described by a CSTR model.

This assured that dissolved contaminants (such as fluoride) that may be brought to the WTP from the river were efficiently diluted by the lagooning basin before entering the WTP.