Numerical substantiation of the parameters of the air-hydraulic hood by a diaphragm

The article is about calculating the main parameters of air hydraulic hoods with a diaphragm to reduce the emergency consequences of a water hammer, possible in the pressure pipelines of an irrigation pumping station. Based on the results of numerical studies by the method of finite differences of the proposed hydraulic shock absorber, dependencies were obtained based on a certain air in the absorbers, the total capacity of the cylindrical cap was determined to determine the main dimensions of the absorbers. Based on the results of numerical studies by the method of finite differences of the proposed hydraulic shock absorber, dependencies were obtained based on a certain air in the absorbers, the total capacity of the cylindrical cap was determined to determine the main dimensions of the absorbers. To determine the economic dimensions of the proposed cap design, comparative calculations of numerical experiments with experimental data prove the reliability of the proposed dependencies using the finite difference method.


Introduction
In the event of a sudden power outage of pumping stations, hydraulic shocks occur in pressure water lines from a decrease in pressure.
AHC is a steel vessel with a cylindrical shape, installed on a pressure pipeline and filled in the upper part with air.
The advantages of AHC include simplicity of design, trouble-free operation, elimination of water discharge, and the absence of a possible rarefaction in the pressure pipeline. With correctly found geometric dimensions of the cap, a high degree of damping of the maximum heads during hydraulic shock (HS) is guaranteed [8][9][10].
A.F. Mostovskiy [1] considers HS in horizontal pressure pipelines when installing AHC at the end of the pipeline in front of the valve. Based on the application of the equation of "living forces", the author obtains formulas for calculating the maximum pressure increase ∆Р for a known volume of the bell. In this case, the author considers the elasticity of the pipeline walls and the compressibility of water and air.
L. Bergeron in [2] gives a graphical method for calculating pressure pipelines for hydraulic shock in the presence of an air-hydraulic cap. This calculation method is characterized with high accuracy.
N.E. Jukovskiy [3] proposed calculating the HS in the presence of a AHC installed on the pipeline. At the same time, the author proposes 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 process of HSh is fast [3]. I.A. Charny [4] uses linearized HSh equations to calculate AHC. In this case, the author adopts an isothermal law (n = 1.0) for the compression and expansion of air in the AHC.
In practice, the most widespread is the calculation method proposed by Evangelisti [5]. Evangelisti's method [5] is based on the use of special graphs compiled to approximate integration of the water hammer wave equations by the finite difference method.
The disadvantage of these graphs is the limited range of variation of the initial parameters, and therefore in many cases, the Evangelisti method is not applicable.
V.S. Dikarevsky in [6], trying to eliminate the drawback of the G. Evangelisti [7] developed an analytical method for sizing a bell located at the end of a pressure line in front of a gate valve. In this case, the author accepts the isothermal law (n = 1.0), the 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.
B.F. Lyamaev in work [8], developed a method for calculating AHC on electronic computing machines (ECM). The proposed method is based on the joint solution of the equations of HS, continuity in the node connecting the cap to the pipeline and the state of the gas in the AHC. The calculation is performed by the author [8] by the iteration method. D.A. Fox gives in [9] a numerical method for calculating AHC. The author applies the method of characteristics with a regular rectangular grid with constant steps ∆х and ∆t. The author jointly solves the equations of continuity, the state of air (gas) and the equations of relations on the characteristics. The calculations are carried out on a computer ECM. The author [9] proposed to take the value of the polytropic coefficient equal to n = 1.20 in the calculations and take into account the pressure loss along the length according to the hypothesis of quasi-stationarity. To damp the intensity of water hammer in the pressure pipelines of irrigation pumping stations, an air-hydraulic cap with a diaphragm is used (Fig.  1); the dimensions of the cap are determined according to the conditions for starting and stopping the pumping unit [10]. In this work and Fig. 1, we use the notation that we described earlier in detail in [10].

Methods
The method for calculating the air-hydraulic hood with a diaphragm installed at the beginning of the pressure pipeline [10] (Fig. 1) is presented.
The proposed method assumes that the pump is turned off instantly, and the time for closing the check valve is zero [10].
To solve the problem of water hammer in the system air -hydraulic cap -pipelinereservoir, the following wave equations of hyperbolic type are used [6,8,10]: Equations (1), (2) and (3) are solved under the following initial conditions [6,8,10]: By analogy with Evangelisti, we will solve the system of equations (1) -(3) by the method of finite differences [10].
Following equation (3) Following equation (2), we have where t  is the value  at time t.
Since W  by (5) where do we get When deriving the calculated dependencies (7) and (10) The dimensionless head h in the pipeline at the place where the cap is installed differs from the dimensionless head h in the cap by the amount of head loss in the "diaphragm", namely [10] до h h h      (12) Using dependencies (7), (10) and (12) AHC with a diaphragm is the most promising design, since here, during a transient oscillatory process in the AHC system -pressure pipeline-reservoir (see Fig. 1), independent regulation of the direction of fluid movement is achieved, that is, one of the main disadvantages of AHC of unilateral action is eliminated [5,6,10].
The main idea in using AHC with a diaphragm is to maximize the use of the effective qualities of AHC both in the phase of decreasing pressure and in the phase of increasing pressure [10].
The problem is posed of determining the minimum absolute head Н аmin and the maximum absolute head Н аmax in the bell in the first oscillation period.

Results and Discussion
When solving the problem, the following assumptions were made: the pump unit turns off instantly; the closing time of the check valve is zero; the problem is solved based on the socalled "rigid" model of unsteady pressure fluid movement, which does not consider the elastic properties of the fluid and the walls of the pipeline [5,10].
Let's write down the main calculated dependences of the AHC with a diaphragm Analysis shows that dimensionless heads h min and h max are functions of the parameters σ, c, д h and n, that is, h ( max Z ) using dependencies (13) and (14) In all calculations, the time step Δ was taken to be the same and equal to t = 0,01. To check the reliability of the above numerical calculations of the AHC with a diaphragm, experimental studies were carried out [10]. A comparison of numerical experiments and experimental data on the study of AHC with a diaphragm is shown in Fig.  2.

Conclusions
1. Analysis of existing scientific works shows that in the event of a sudden power failure in the pressure systems of pumping stations, a HSh appears with a discontinuity of the flow, which negatively affects the normal operation of the pumping unit. To prevent this phenomenon, it is very important to develop a new method for calculating the optimal dimensions of the AHC with a diaphragm. 2. With optimal damping of the maximum pressure of the main unit in the pressure pipelines of pumping stations, it is very important to take into account the changes in the local resistance in the diaphragm of the connecting pipeline and the law of expansion and contraction of the air of the proposed bell design. These factors must be considered when integrating the wave equations of hyperbolic type HSh by the numerical method -the method of finite differences to establish the optimal parameters of the proposed cap design. 3. As a result of the numerical solution of the differential equations of the HSh, dependencies are proposed for calculating the economic dimensions of the proposed design of the HSh damper. The reliability of the above numerical method is proved by comparing the values of σ number and with the experimental values of σ оp [10].