# MATHEMATICAL AND COMPUTER MODELING OF PUMPS IMPELLERS FOR IMPLEMENTATION ON A 3D-PRINTER

### Abstract

Centrifugal pumps are one of the most common pumps in the world. Centrifugal pumps are so versatile that they are used in almost all industries (chemical, mining, food, energy, etc.). The main advantages of centrifugal pumps are simple design and high efficiency. Among the shortcomings worth noting is the low wear resistance of the impeller. The shape and size of the pump impeller blades are determined by time-consuming calculations. Pump impellers can be divided into three categories: open, semi-closed, closed.

Various manufacturing methods are known for centrifugal pumps impellers. The processing of the blade disk profile can be done on a program-controlled copy-milling machine, cover disc and blade disk are welded to each other by argon-arc welding, and the wheel is heat-treated to relieve stresses after welding. The disadvantages of this method are the difficulty of ensuring high wheel strength. In this case, the welds distort the shape of the inter-blade channels.

Sometimes the impeller is made of two parts, forming each blade by means of a butt weld joint of two elements. The disadvantages of this method are technological difficulty, and in some cases the impossibility of obtaining a complex spatial shape of the inter-blade channels for small wheels.

In foreign and local pump engineering, researches continue to be carried out to find optimal process flow schemes for preparing parts from polymers. The main methods of plastic products processing include pressing, injection molding, extrusion, vacuum forming, blowing, stamping.

The global manufacturer of various types of pumps and valves, the KSB concern (Germany), has become the world’s largest innovator in the field of 3D printing applications in the foundry industry. With the help of a 3D printer, master models and spare parts for pumps are manufactured, which can significantly reduce the time to repair or manufacture new models.

However, there is the problem of setting information for printing, i.e. creating a mathematical and computer model of the designed object.One of the methods for solving this problem is the application of the R-functions theory, which allows one to describe geometric objects of complex shape with a single analytical expression.

The aim of this work is to develop, on the basis of the theory of R-functions, methods of mathematical and computer simulation of pump impellers for their implementation on a 3D-printer.

When constructing mathematical models using the R-functions method, both the simplest R-operations as ; and R-operations for smoothing sharp edges and corners ; will be used. When constructing equations corresponding to geometrical objects with cyclic-type point symmetry, to reduce the number of R-operations a method based on the theorem proved earlier was developed.

To construct an object of a given thickness, its normalized equation was used, which also in some cases allows to significantly reduce the number of R-functions.

Several mathematical models of open impellers in centrifugal pumps were built: with radial blades, with six curved blades and balancing holes and with five curvilinear blades of different inclination angles in the central zone. By varying the alphabetic parameters entered into the program, impellers with other structural elements (such as number of radial blades, number of balancing holes, various blade inclination angles in the central zone, etc.) may be obtained within the same program. The obtained equations are visualized using RFPreview program, and their 3D-printer implementation is made in the Institute of Mechanical Engineering Problems of the National Academy of Sciences of Ukraine.

On the basis of the developed algorithms, mathematical and computer models of semi-closed impellers of centrifugal pumps were built.

The advantages of using 3D printing are obvious – the production of non-standard models, the reduction of time for the creation of new prototypes and the reduction of repair time, the simplicity and considerable cheapness of production, the use of modern heavy-duty materials.In this paper, the theory of R-functions is used for mathematical and computer simulation of pump impellers in the implementation of 3D printing technology.The method of constructing equations of geometrical objects with cyclic symmetry type in 3D was used to construct the equation of an open centrifugal pump impeller with radial blades.And to build mathematical models of the centrifugal pumps impellers with curved blades, a new approach was developed based on the theory of R-functions.

Analytical recording of the designed objects makes it possible to use alphabetic geometric parameters, complex superpositions of functions, which, in turn, allows to change their structural elements quickly.

### References

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*Bulletin of Zaporizhzhia National University. Physical and Mathematical Sciences*, (2), 166-176. Retrieved from http://journalsofznu.zp.ua/index.php/phys-math/article/view/267