Optimal volume of flow tanks for rainwater management

. The purpose of the article is to specify and supplement the methodology of engineering calculation of rain runoff control at the treatment plant, given the current regulatory and methodological framework. Two flow management schemes, including an intermediate tank, a pumping station, and a separation chamber, have been analyzed by mathematical modeling. Changes in the calculation formula for determining the required volume for the tank connection scheme when emptying the tank by gravity are proposed. There is a significant difference in the calculation results according to the proposed and current regulatory methodologies.


Introduction
Surface, including rain and snowmelt, untreated wastewater causes serious environmental and health problems, being the main source of pollution and a huge mass of waste that is washed away from the territory of industrial facilities and residential buildings.This ultimately destroys the environment and endangers the ecological safety of the population [1][2][3].Underground pipeline systems and treatment plants, usually of a storage type, are used for the orderly collection, disposal, and purification of wastewater from pollution [4][5][6].
The installation of management (storage) tanks on surface runoff networks is mainly carried out in front of treatment facilities in order to reduce the volume of runoff.Intermediate storage tanks perform several functions.
Firstly, they help prevent overloading of the water treatment system, which can occur during heavy rains or floods [7][8][9].During the peak flow period, rain runoff temporarily accumulates in the tanks, and when the storm flow decreases to a certain value, emptying occurs.In addition, the accumulation in the regulating tanks allows the deposition of coarse impurities before they enter the treatment system [10][11].
In practice [4] there are three main schemes for placing tanks in the surface runoff drainage system.In this paper, two of them are considered in Figure 1.
The first scheme includes a separation chamber, which redirects the rain flow with a certain flow rate exceeding the limit value to the tank.This avoids overflow of the collector and overload of the treatment plant.To empty the tank, a pumping station with a constant supply is used, which does not exceed the calculated flow rate in the network after the separation chamber.The second scheme is the simplest of the recommended ones and assumes the presence of a bottom tray in the regulating tank, the discharge capacity of which should be equal to the capacity of the outlet pipeline.When the flow rate exceeds the limit value, the water fills the tray and is poured into the storage tank [12][13].
When the entire flow of rainwater enters the tank with its simultaneous emptying through a small-diameter pipeline (flow tanks, scheme 2 in Fig. 1), the flow management and volume of the tank will differ from those recommended in the regulations [4].
The task of this work is to clarify the dependencies for determining the design parameters of the selected control schemes, taking into account the inflow and outflow regimes.

Materials and methods
The volume of regulated runoff W st and the depth of water Н in the flow tank can be found by jointly considering the equations characterizing the flow of rainwater into the tank and the outflow of water from it.The "rational method" and the power function are used to calculate the intensity of rain based on how long it lasts [14][15][16][17].
For calculating rains with a maximum intensity at the beginning and with a uniform increase in the flow area along the length of the network, the change in flow rate along the course of rain on the ascending Q' in and descending Q" in branches of the flow hydrograph in Fig. 2 is expressed by the dependencies:  during the period t ≤ t r : (1)  during the period t r < t: Where t r is the estimated duration of rainwater flow to the storage tank; n is a parameter characterizing the intensity and duration of rain, accepted according to [4,18] depending on the region's geographical location.
The flow rate of the flowing water Q out can be determined by the formula for the outflow of liquid through a flooded hole:

Results
In the absence of leakage (Q out = 0) and a constant area of the water mirror in the tank S st at different values H st , the water depth changes according to the dependencies:  during the period t ≤ t r : (4)  during the period t r < t: (5) Taking into account the simultaneous flow of the drain, the water depth in the tank will be equal to: (6) The volume of water flowing out during this time t will be: (7) Since H st = f(t), the joint solution of equations ( 3), ( 5) and ( 7) can be performed by iteration.For example, for time t ≤ t r , after integrating equation (7), we obtain: Taking initially H st = H in and substituting the obtained value of the depth of H in into the dependence (7), we obtain at = const: When substituting H st into expression (7), we get a new value (H" st ), which will be slightly larger than H in , i.e.H in > H" st > H' st .
In real conditions, the area of tank S significantly (by several orders of magnitude) exceeds the cross-ec al area f he d charge p pe ; heref re, w h b eq e substitutions of the refined values H in equation ( 7), the volume W out and therefore the depth H st practically do not change.
For t > t r , the solution of equations ( 5) and ( 7) can be carried out by numerical or other approximate methods.In the first approximation, the H st values can be taken to be equal to: The highest depth of liquid in the tank H max will correspond to the time t in determined by the iteration method, at which the incoming flow rate determined by the dependence (2) will be equal to the value Q out calculated by the formula (3).The estimated volume of the intermediate tank will be: The value of , or the diameter of the discharge pipe D out , should be determined by the equations (3), ( 4) and ( 9), provided that Q out = α r Q 0 at при t = t r .Computing by formulas (3), (5), and (7), performed by the iteration method, requires a large number of calculations.
At the same time, the difference in the value of W st , as a rule, does not exceed 5-7% compared to calculations using formulas (7) and (12), and simplified dependencies can be obtained.
Formula analysis (3-5), ( 9) and ( 12) allowed us to take with sufficient accuracy for practice the following ratios between the costs flowing from the tank in different periods:  during the period t ≤ t r (14) E3S Web of Conferences 463, 02011 (2023) EESTE2023 https://doi.org/10.1051/e3sconf/202346302011 during the period t r < t (15) The volume of the intermediate storage tank in this case, taking into account dependencies (1) and ( 2) for determining the flow rate entering the tank, will be: The time value t in will depend on the accepted reg la c eff c e α r , the nature of the inflow of rainwater into the tank, and the time at which Q in = Q out .Taking into account ( 2) and (15), the relationship between time t in a d c eff c e α r is expressed by the formula:

Discussion
To integrate equation ( 16), mathematical modeling methods using a universal CAS (computer algebra system) were used [19,20].After some simplifications, the formula for determining the volume of the storage tank when it is connected to the rain sewer network according to scheme 2 in Figure 1 is given as: (18) Where k st is the volume coefficient of the storage tank expressed in terms of the maximum flow of rainwater: (19) For example, the values of coefficients k st calculated by formulas ( 17) and ( 19) at n = 0.75 w ll be: a α r = 0.1, k st = 0.99; at α r = 0.5, k st = 0.50.
Thus, the difference in the values of k st , and hence W st , can reach 50% or more.The volumes of the flow tanks, scheme 2 in Figure 1, should be determined by formulas (18) and (19).

Conclusion
Thus, the following results were obtained in the paper: E3S Web of Conferences 463, 02011 (2023) EESTE2023 https://doi.org/10.1051/e3sconf/202346302011 A calculation formula is proposed for calculating the volume of the storage tank when it is connected using a gravity flow scheme. The formula for calculating the volume coefficient of the intermediate storage tank is substantiated, which depends on the regulation coefficient, the location region, and the duration of the flow of rainwater to the tank. It is established that the volume of the tank calculated according to the proposed method may differ from the recommended volume in regulatory literature by 50% or more.

( 3 )
E3S Web of Conferences 463, 02011 (2023) EESTE2023 https://doi.org/10.1051/e3sconf/202346302011WhereH is the pressure above the center of gravity of the discharge pipeline (the water dep h he a k); s he area f he fl ded pe g ( he e ra ce he p pel e), a d is the flow coefficient.

Fig. 2 .
Fig. 2. Calculation schemes for determining the storage tank volume. 1 -flow hydrograph, 2 -flow rates in the collector after the separation chamber.