Analytical substantiation of the parameters of the directional air-hydraulic hood

The article is about to the calculation of the main technical parameters of the directional air-hydraulic hood. It is accepted as a mathematical model of wave equations of hyperbolic type as applied to pressure pipelines of pumping stations, and analytical solutions of wave equations for different values of the polytropic exponents are proposed. Water hammer poses a danger to the normal operation of the main equipment of stations, control and measuring equipment, control devices and pressure pipelines. To damp the intensity of water hammer in the pressure pipelines of pumping stations, we have accepted an effective design of an air-hydraulic cap. When establishing the strength indicators of pressure pipelines against hydraulic shock, it is necessary to make an accurate calculation of the main parameters of the proposed design of the air hydraulic cap. The article presents the results of analytical and experimental studies of the accepted cap design. At the same time, an analytical method is proposed for calculating the basic dimensions of the cap. The results of the proposed calculation procedure are in good corresponding with the experimental data. This confirms the reliability of the proposed analytical calculation method.


Introduction
To protect pressure pipeline systems from the impact of hydraulic shock, various dampers are used, in particular, air-hydraulic caps (AHC) .
In the work of N. E. Zhukovsky [1], a method for calculating the hydraulic shock (HSh) in the presence of an AHC installed on the pipeline was proposed. At the same time, the author offers an approximate formula for determining the volume of air in the AHC and accepts the adiabatic law of compression and expansion of air in the AHC, since, according to the author, the HSh process is fast-flowing [1]. I.A. Charny [2] uses the linearized HSh equations to calculate the AHC. In this case, the author accepts the isothermal law (n = 1.0) for the compression and expansion of air in the AHC.
In practice, the calculation method proposed by Evangelisti [3] is the most widely used. This method is based on the application of special graphs compiled by Evangelisti due to the approximate integration of the wave differential equations of hydraulic shock by the finite difference method.
The disadvantage of these graphs is the limited range of changes in the initial parameters, and therefore, in many cases, the Evangelisti method is not applicable.
In V. S. Dikarevsky's work [4], he tried to eliminate the drawback of the G. Evangelisti for the isothermal law (n = 1.0) in a wide range of changes in the parameters σ and тро h .
However, the author admits inaccuracies in solving the basic equations F.M.Darson and A. A. Kaliske [5] presented an analytical method for determining the size of the cap located at the end of the pressure line in front of the gate valve. At the same time, the author accepted the isothermal law (n = 1.0) change in the volume of gas in the AHC and does not consider the effect of pressure losses on friction in the pressure pipeline. This method of calculating the cap is also approximate.
In the work of B. F. Lyamaeva [6], a method for calculating the AHC on a computer is developed. The proposed method is based on the joint solution of the equations of HSh, continuity at the junction of the cap to the pipeline, and the state of the gas in the AHC. The calculation is made by the author using the iteration method.
In the work of D.A.Fox [7], a numerical method for calculating the AHC is given. The author applies the method of characteristics with a regular rectangular grid with constant steps ∆x and ∆t. The author solves the equations of continuity, the state of air (gas) and the equations of relations on the characteristics. The calculations are implemented on a computer. The author [7] suggests taking the polytrope coefficient value equal to n = 1,20 and considering the head loss along the length according to the quasi-stationarity hypothesis.
To extinguish the hydraulic shock that occurs in pressure pipelines, along with other shock dampers, a directional air-hydraulic cap (AHC) is used ( Figure 1); the size of the cap is determined by the conditions for starting and stopping pumps [2][3][4][5][6][7][8][9]. In this paper and Figure 1, the notation described earlier [9] is used in detail.

Methods
This paper illustrates a calculation method directed to the AHC installed at the beginning of the pressure pipeline ( Figure 1).
To calculate the hydraulic shock in the system of the AHC -pipeline-reservoir, the following system of equations is used [4,9]: The system of equations (1) is solved under the following initial conditions: (2)

Results and Discussion
To determine Н аmin and Н аmax in a pressure system with a directed AHC, the following dependencies are obtained as a result of the analytical solution of the system of equations.
To determine Н аmin a) at n ≠ 1  To determine Н аmax а) at n ≠ 1     It is also determined that the formula (11) can be applied to perform preliminary calculations of the directed AHC in the following range of changes in the parameters It should be mentioned that the use of approximate formulas (11) and (13) allows us to determine the volume of air in the AHC with a margin of 0...15 % [9]. From equation (6) or (7), for known σ, m 2 , and h min , the values of the quantity can be determined by the iteration method ζ max , and then calculated The problem of determining ζ max , h max and H amax can be solved on a computer. To verify the reliability of the above numerical calculations of the AHC with a diaphragm, experimental studies were conducted. The results of the comparison of the calculated formulas (11) and (13) and the experimental data [9] for the study of directed AHC are shown in Fig.2.   Fig. 2. Comparison of the results of analytical calculations using formulas (13) and (15) of directed AHC with experimental data [9].

Conclusions
The analysis of literature sources presents that in the event of a sudden power outage in the pressure pipelines of irrigation pumping stations, a dangerous non-stationary process occurs, a special case of which is the HSh. To prevent this phenomenon, it is very important to develop a new method for calculating the optimal size of the HSh -directed AHC extinguisher.
To optimally dampen the intensity of HSh in the pressure system of pumping stations, it is imperative to take into account changes in the local resistance in the diaphragm of the connecting pipes and the law of expansion and compression of air in the proposed design of the hood. These factors must be considered when integrating the wave differential equations of the shock wave by the finite difference method to determine the optimal dimensions of the proposed cap design.
As a result of the analytical solution of the wave differential equations of the HSh, the dependences for calculating the economic parameters of the proposed design of the HSh extinguisher are offered.
The reliability of the proposed formulas (11) and (13) is proved by comparing the calculated values of σ calc and with the experimental values of σ OP [9].