A simplified dimensioning method for high-efficiency retention tanks

. The article presents the results of research on the development of a simplified dimensioning method for sewage retention tanks equipped with a high-efficiency hydraulic system. The need to develop a new method is associated with the outdated rainfall model by Błaszczyk, which is the most popular model in Poland. According to the research made by numerous scientific centres, this model underrates the values of the maximum intensity of critical rains. As a consequence, its use affects the design of sewage systems and related facilities with insufficient hydraulic capacity.


Introduction
Urban spatial development is accompanied by problems related to the implementation of water and sewage management [1,2]. Among these problems, the most important include the formation of so-called local municipal floods [3] and a lowered quality of sewage receiving bodies.
Municipal floods are very often caused by the insufficient hydraulic capacity of drainage systems, and generate significant material losses. In turn, the contamination of receiving bodies is most often associated with the discharge of a significant volume of sewage containing a large load of organic and inorganic compounds in a short time [4]. In both cases, the main stimulator is an increase in the volume necessary to discharge rainwater, which, among others, is caused by the sealing of the drainage area and the intensification of torrential rains.
Limiting or even the complete elimination of the negative effects of the presented problems can be obtained by using sewage retention facilities [5,6]. Their ability to temporarily deposit excess liquid allows reducing the flow of sewage in buildings located below the tank to a level declared by the designer. This beneficial effect has made retention facilities an integral part of many urban investments [7,8]. At the same time, it should be noted that every capital-intensive investment is required to execute a proper financial analysis [9,10,11].
On the other hand, the use of retention facilities requires a suitably adapted method to determine their necessary capacity. Nowadays, the most favourable solution seems to be the use of hydrodynamic modelling software for this purpose. Unfortunately, this requires dedicating a significant amount of work for the construction and calibration of the model, which in the case of small investments is unfavourable due to significant capital intensity. In such cases, a good alternative to achieve similar effects is the use of simplified methods based on the use of simple algebraic formulas.
The methods that are currently used in Poland for simplified dimensioning of tanks are based on the model rainfall by Błaszczyk [12], which is unreliable and should be replaced by another rainfall model, for example, by Bogdanowicz and Stachy [13]. Therefore, due to the update of rainfall models in Poland, the simplified dimensioning methods of retention tanks should also be updated.

Aim of the research
The aim of this manuscript is to develop calculation formulas that would allow determining the capacity of a retention tank equipped with the installation of retention facilities [6]. When determining them, the most current rainfall model by Bogdanowicz and Stachy for Poland was used [13].

Installation of retention facilities
The installation of retention facilities is a system of hydraulically connected ducts, the installation of which in any retention facility allows reducing the time necessary to obtain full hydraulic capacity of a building's outflow channel.
The application of the installation of retention facilities in classic retention tanks (Fig. 1a) allows obtaining the value of the volume stream of sewage that flows outfrom the tank that is close to the flow rate of sewage flowing to it already in the initial phase of its functioning. To illustrate this phenomenon, Figure 1b presents sewage inflow and outflow hydrographs for a tank equipped with the installation in question. As a result, the use of this solution translates into a reduction in the required retention capacity of the facility.

Selected rainfall models for Poland
The most popular formula in Poland that describes the intensity of rain q depending on its time t, the average annual height H, and frequency c is Błaszczyk's formula [12] described by formula (4.1). where: H -average annual height, mm; c -frequency of rain, 1/c years; t -reliable time of rainfall for a network, min. However, this rainfall model has been deeply criticized in recent years. As research shows, it significantly reduces the intensity of rainfall, which is why it should no longer be used in the dimensioning of drainage systems in this form [14]. The probabilistic rainfall model by Bogdanowicz and Stachy [13], which is an alternative to Błaszczyk's formula [12], covers almost the whole of Poland. Depending on the assumed probability of rainfall p and the geographical region of Poland R, at a specific rainfall time top it allows determining the total rainfall height hmax according to the dependence (4.2).
where: hmaxthe maximum total rainfall height of a determined time top and exceedance probability p, mm; prainfall exceedance probability; α (R , top ) -parameter (scale) depending on the region of Poland R and time of rain top. To use the presented formula (3.2), the region of Poland R for which the rainfall model is calculated should be designated first. The boundaries of Poland's division into regions: R1, R2, and R3 are presented in the works [15]. Whereas the scale parameter α, due to Poland's division into three regions, was described by six different formulas R and rainfall time top.
Research on the suitability analysis of this rainfall model for sewage system design was made by [15]. It was pointed out that although for rains with the frequency of c equal to one, this model is subject to a significant error but it is currently one of the most suitable models for drainage system design [15].

Dimensioning of retention facilities using the formula by Bogdanowicz and Stachy
The development of the dimensioning method of retention tanks equipped with the installation of retention facilities was based on the basic dependence describing the volume of sewage necessary to be deposited. This dependence is presented by formulas (5.1 and 5.2) and they show that the necessary retention capacity of a facility is equal to the surface area between the sewage inflow and outflow hydrographs [16].

=
(5.1) where: A -time-variable inflow of sewage to the tank, l/s; B -time-variable outflow of sewage from the tank, l/s; tdtime after which the flow of sewage into the tank decreases to the level of the outflow intensity, min. For tanks equipped with the installation in question, sewage inflow function A coincides with outflow function B to the value of the maximum sewage outflow from the tank (Fig.  1a).
In the development of simplified formulas, a simplified hydrograph of a trapezoidal sewage inflow was adopted, the lower base of which is equal to the time of rainfall authoritative for tank Tz increased by an authoritative time of rainfall for network ts. In turn, its upper base corresponds to the time of rainfall authoritative for tank Tz decreased by the time of rainfall for network ts (Fig. 2). Thus, the required capacity of the facility V corresponds to the surface area of the figure between the maximum value of sewage inflow to the tank (straight A1), and the determined value of sewage outflow from the tank (straight B1). The developed simplified dimensioning method of retention tanks is based on the rainfall model by Bogdanowicz and Stachy. However, its form is heterogeneous because it makes its form dependent on the time of rainfall t and the region of Poland R. Hence there is a need to develop several different calculation formulas for individual regions and times of rainfall. To systematize them, they were given appropriate markings and are presented in Table 1. The procedure for determining individual calculation formulas is presented using the example of formula Ia. The remaining equations were determined in an analogous manner.
In accordance with the formulas (5.1 and 5.2), the necessary retention capacity corresponds to the surface area between the hydrograph of inflow A and outflow B (Fig. 3). Therefore, using the installation of retention facilities, the only task becomes determining the trapezoidal surface area above line B1 (Fig. 2). For this purpose, the phenomenon of the similarity of triangles as in the work by [16] was used. The surface area of the desired figure can be divided into three simple geometrical figures: two identical rectangular triangles P1, and rectangle P2 (Fig. 3).
Triangle P1 is characterized by a base length of x and a height that is equal to the difference between the maximum values of the inflow and outflow of sewage A-B.Whereas, rectangle P2 has a height equal to the difference between A -B and a base with a length equal to the difference of time of rainfall for the dimensioning of the retention tank Tz and time of rainfall for the dimensioning of network ts.  In turn, the surface areas of components P1 and P2 can be described using formulas (5.4 and 5.5).  The last step was to determine the value of maximum inflow A. It depends on the rainfall intensity value and the area of the reduced drainage area. After substituting all values and adopting calculation formulas for the intensity of rainfall by Bogdanowicz and Stachy, six calculation formulas (5.8-5.13) were obtained for individual regions of Poland and time of rainfall t according to Table 1 where: Tz -time of rain for retention tank dimensioning, s; b -maximum intensity of sewage outflow from the tank, m 3 /s; ts -time of rainfall for network dimensioning, s; p -rainfall exceedance probability; ∑F -reduced surface area of drainage area being drained, ha. The presented calculation formulas (5.8 -5.13) allow determining the required capacity of retention tank V for a given length of rainfall time t. However, it will vary depending on the rainfall time for tank Tz adopted for the calculations. Therefore, to determine the required tank capacity, calculations should be made for different rainfall times Tz and the highest value should be chosen. This is presented in the example for the data: reduced surface area of drainage area ∑Fzr -25 ha, rainfall time authoritative for the network of Ts -15 minutes, probability of rainfall p -0.2, the maximum sewage outflow intensity from tank B -0,5 m 3 /s.
The presented data was introduced into calculation formula Ia (4.8), and then the required retention capacities V for rainfall times Tz from 0 up to 100 minutes were determined. The results obtained are shown in Figure 4. The presented data (Figure 3) shows that the largest required retention capacity of tank V amounting to 5 207 m 3 was obtained for a rainfall time of Tz 70 minutes.
The presented methodology for the dimensioning of retention tanks equipped with installations for retention facilities has been made for design cases in which the reliable time for the dimensioning of retention tanks Tzis longer than the time for dimensioning of network ts.