# Assessment of Changes in the Abrasiveness of Solid Particles in Hydraulic Mixtures Pumped with ESPs
## 2 Materials and Methods
Quantitatively, the value of the abrasion index, \(AI\), is determined with Equation (3):  
\[AI=10\cdot(2-K_{r}-K_{s})+0.3\cdot M_{-0.25\%}+0.25\cdot\left(M_{i\%}+M_{q \%}\right) \tag{3}\]  
where \(M_{-0.25\%}\) is the mass fraction of abrasive particles of mechanical impurities with a size of less than 0.25 mm,%; \(K_{s}\) is the particle-shape coefficient of the mechanical impurities, found according to the API method; \(K_{s}\) is the sphericity coefficient of the mechanical-impurity particles, found according to the API method; \(M_{i\%}\) is the mass fraction of insoluble residue (particles insoluble in acid),%; and \(M_{q\%}\) is the fraction of quartz particles in the total mechanical-impurity mass,% [7]. Analysis of formula (3) shows that the largest influence on abrasiveness of particles of mechanical impurities that interact with parts of electric centrifugal pumps is made by the sizes and shapes of solid particles: their roundness and sphericality. A diagram for visual determination of the values of the coefficients of roundness, \(K_{r}\), and sphericality, \(K_{s}\), of particulate mechanical impurities is shown in Figure 2[22]. Most of the methods used to protect submersible electric centrifugal pumps from hydroabrasive wear are aimed at reducing the concentration and average particle size of mechanical impurities that pass through the pump stages in pumped liquid flow, increasing the abrasion resistance of pump stage materials and improving interstage seals [23]. The shapes and sizes of the solid particles contained in a liquid stream change as they pass through the stages of centrifugal pumps; mechanical-impurity particles are ground, while the values of the circularity and sphericality coefficients, \(K_{0}\) and \(K_{s}\), respectively, of the particles change [24; 25; 26]. In accordance with [PERSON]'s law, when a rock particle is crushed with mass, \(m_{0}\), from the initial diameter, \(D_{i}\), to the final diameter, \(D_{f}\), energy is determined with Equation (4):  
\[E_{cr}=3.6\cdot k_{p}\cdot k_{s}\cdot k_{c}\cdot\left(\frac{1}{\sqrt{D_{f}}}- \frac{1}{\sqrt{D_{i}}}\right)\cdot m_{\rm i} \tag{4}\]  
where \(E_{cr}\) is the energy required for crushing a given volume of rock, kJ; \(k_{p}\) is the proportionality coefficient of specific energy consumption for formation of new surfaces during destruction (Bond coefficient), kWh\(\cdot\)m\({}^{0.5}\)/t; \(k_{s}\) is the particle-shape coefficient; \(k_{c}\) is the coefficient of initial attenuation in particles (grain boundaries in multicomponent particles, cracks of natural and man-made origin, etc. ); and \(m_{\rm i}\) is the initial mass of a crushed particle, kg. Upon contact with the metal of the working stage, a solid particle moving in a liquid flow will be transferred to energy, \(E\), the value of which is determined with Equation (5):  
\[E=0.5\cdot v^{2}\cdot m_{\rm i} \tag{5}\]  
where \(v\) is the speed of movement of the solid particle in case of impact with the working stage, m/s. Decreases in the diameter and mass of a solid abrasive particle cause a decrease in the value of the energy, \(E\), transmitted to the particle when it hits the metal of the working stage of the centrifugal pump. At \(E<E_{cr}\), solid particles of mechanical impurities cease to grind [22; 27]. The passage of mechanical impurities contained in a pumped-liquid flow through the pump stages determines the change in the abrasiveness of the solid particles [28]. At the same time, the wear process has implicit features characteristic of various sections of a multistage pump [29; 30; 31]. The identification and study of these features of the hydroabrasive-wear process of the stages of electric centrifugal pumps is most simply carried out during laboratory experiments on special test benches [32; 33].