Mathematical model of the pumping unit of machine water lifting systems

Pumping stations have been investigated as objects of control and energy saving. Methods for determining energy-efficient processes of functioning of water-lifting pumping stations with the formation of their energy-saving modes are given. Mathematical methods have been developed for describing and modeling a pumpingunit for machine water lifting systems.


Introduction
Nowadays, one of the main sources of meeting the growing demand of the economy of the Republic of Uzbekistan for electricity should be energy and resource conservation. Therefore, the tasks of developing energysaving technologies in the sectors of the economy of the republic, in particular, at such energy-intensive objects as irrigation pumping stations of machine water lifting systems acquire a decisive importance. It is known that the main technical and economic indicators of the operation of a pumping unit as indeed of the entire pumping station are the efficiency of the pumping unit (station) and the specific power consumption spent on the supply (pumping) of a unit of water volume into the hydraulic pressure network [1 ]. It was determined [2] that when forming a mathematical model of a pumping unit for machine water lifting systems to ensure energysaving modes of operation, as a criterion for assessing the efficiency of its work, it is necessary to take the minimum specific consumption of electrical energy for water supply with appropriate restrictions imposed on the conditions of its operation. Due to the fact that in the overwhelming majority of pumping stations of machine water lifting systems pumping units are operated as a group of pumping units structurally combined to work together in a common pressure pipeline, the value of the specific electricity consumption for pumping units is initially determined.

Experimental research
As is known [3], the specific power consumption for the supply of 1 million m3 of pumped water to a height of 1 meter of water column for pumping units is determined by the following formula: There ߟ -efficiency of ݅ -th pump unit as part of a pumping unit; i K -pump efficiency of݅-th pump unit; ߟ . -efficiency of the drive motor݅-th pump unit; ߟ ். -transmission efficiency. The current value of the efficiency of the pump by taking into account the cavitation-abrasive wear of its working bodies can be determined by the expression [4]: where ‫ܭ‬ ௪ -the wear factor of the pump's working bodies, calculated as ‫ܭ‬ ௪ =е -0.00000833*Т ; Т-the duration of the pumping unit; η -the value of the efficiency of the pump determined for each specific operating condition of the pump unit in accordance with [5]. The value of the efficiency of the pump drive motor by taking into account its load can be determined by the formula: where ߟ . -therated value efficiency of the drive motor; А ் -loss factor of the electric motor. For an asynchronous motor: at݊ ≤ 1000 RPMА ் =0.5, at݊ ≥ 1000 RPMА ் =0.7; and for a synchronous motor: at݊ ≤ 1000 RPMА ் =1, at݊ ≥ 1,000 RPMА ் =2). -the mechanical power of the pump is determined by the expression (7) [5]: i Q -capacity (performance) of the݅-thpump at the steadystate operating point. The combination of the above formulas makes it possible to calculate using the expression (1) the specific consumption of electrical energy of each of the pumping units, combined for joint operation into a common hydraulic pressure network as a part of a pumping unit. The specific flow rate of the pumping unit of machine water lifting systems as a whole can be determined as: where ߟ -the efficiency of the pumping unit. An expressionߟ can be obtained for any number of operating pumping units, structurally combined for joint parallel operation into a common hydraulic pressure network.
where N -the number of operating pumping units as part of a pumping unit. Thus, the expression (8), obtained on the basis of (9), allowing to calculate the efficiency of the pumping unit depending on the number of jointly functioning pumping units, as well as taking into account the equations that determine the corresponding performance characteristics of the pumps [5] and (1), (4) for the efficiency of the pump and the drive motor of each of the pumping units, together with the restrictions imposed on the parameters of the pumping unit functioning in the form of:

3.Design Studies
Of the constraints listed (10), a more detailed consideration is required to determine the form of the equation for finding the operating point of the pumping units, located on the pressure networkcharacteristic, which undoubtedly has one of the defining values in describing the technological process of water supply of the pumping unit. The position of the operating point of the "pumping unit -pressure network" system corresponds to its material and energy equilibrium and is characterized by the well-known hydraulic equation that determines the required pressure in the pipeline network: ‫ܪ‬ ோ = Н ௌ + ℎ + ℎ + ℎ + ℎ ௦ , ( 11 ) where ℎ -pressure loss in the suction pipeline of pumping units; ℎ -pressure loss in the communication pipeline of pumping units; ℎ -pressure loss in the supply pressure pipeline of pumping units; ℎ ௦ -pressure loss in the general hydraulic pressure network of the pumping unit. To determine losses in pressure pipelines of pumping stations of machine water lifting systems, one should use a number of known hydraulic relationships and reference data [6], as well as the results of experimental studies conducted by F.A. Shevelev [7].
The pressure loss in the suction pipeline of a pumping unit can be determined by the expression: Suction line resistance   (16) and (14), respectively, using the size (in mm) of the inner diameter of the communication pipeline‫ܦ‬ . . Due to the complexity of the hydraulic phenomena occurring in the pressure pipelines of the pumping station of machine water lifting systems, each local resistance is characterized by its own loss factor, which is usually determined empirically or, in some cases, can be calculated from theoretical data. Wherein‫ܭ‬ ெ for each specific pumping unit of machine water lifting systems taking into account the design of the entire complex of the pipeline pressure network and bearing in mind the presence of certain types of local resistances, which, as a rule, are determined from special reference literature [6,7,8].Head loss in the supply pressure pipelineof the݅th pump unit is calculated in the same way as above using the formula: can be expressed through the flow rate ܳ pumping unit in thefollowing form: (22) The head loss in the general hydraulic pressure network of the pumping unit of the machine water lifting systems is expressed by the equation: where ܴ = ‫ܭ‬ ெ + ‫ܭ‬ ைோ + А ோா + ‫ܮ‬ ; (24) ܴ -the resistance of the common pressure network of the pumping unit of the machine water lifting systems; ‫ܮ‬ -the length of the general pressure network of the pumping unit of the machine water lifting systems. When a pumping unit of machine water lifting systems is operating, the operating point of its operation is determined by the total flow rate-pressure characteristic of the pumping unit and the characteristic of the pipeline hydraulic pressure network in accordance with the equality ‫ܪ‬ ௨ = ‫ܪ‬ ோ .This means that at a given operating point, the pressure developed by the pumping unit of machine water lifting systems is equal to the pressure required by the pipeline network in order to ensure the required supply. Therefore, the equation of the regime point, taken into account in the constraints (10), will take the form:

Conclusion
Thus, all the necessary analytical expressions required for the formation of a mathematical model of a pumping unit for machine water lifting systems have been determined in the form of the dependence of the specific consumption of electrical energy on the design, operational and technological parameters of its functioning, which allow to optimize and study the operating modes of the pumping unit to ensure energyresource saving of the system machine water lifting.