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As stated in Teacher Guan Xingfan’s \"Manual of Modern Pump Theory and Design,\" during operation, an axial force acts on the rotor, and this force pulls the rotor to move axially. Therefore, it is necessary to find a way to eliminate or balance this axial force in order for the pump to function properly. The axial force acting on the pump rotor consists of the following components: 1) The axial force resulting from the asymmetry between the front and rear cover plates of the impeller, a force that acts in the direction of the impeller’s inlet ； 2) Axial forces caused by structural factors such as shaft seats and shaft ends, with their direction depending on the specific circumstances ； 3) Axial force caused by the rotor weight (such as in vertical pumps), which is related to the arrangement of the rotor ； 4) Other factors affecting axial force ； 5) Dynamic reaction force; this force points to the rear of the impeller. The main content of this article comes from the KSB website, showing how Europeans understand axial thrust. Composition of axial thrust: Axial thrust is the resultant force of all axial forces (F) acting on the pump rotor, as shown in Figure 1. Figure 1: Axial thrust of a single-stage centrifugal pump. For a single-stage centrifugal pump, the axial thrust acting on the rotor includes: 1) Impeller axial force (F1): It is the difference between the axial pressures on the impeller cover on the discharge side (Fd) and that on the suction side (Fs), i.e., F1 = Fd – Fs. 2) Momentum force (FJ): This is a force that acts continuously on the fluid within a certain volume (refer to the principle of momentum conservation in fluid mechanics); it is calculated as follows: FJ = ρ·Q·ΔVax, where ρ is the density of the fluid being pumped, Q is the pumping flow rate, and ΔVax is the difference between the axial components of the absolute velocities at the inlet and outlet of the impeller. 3) The resultant pressure generated by the static pressures upstream and downstream of the shaft seal at the shaft cross-section Ass, i.e., FWd = AWd·ΔpWd. 4) Special axial forces, such as those that arise when the vortex conditions in the clearance between the impeller and the casing change during the startup of the pump. 5) Other axial forces, such as the rotor weight (FW) on non-horizontal centrifugal pumps or the magnetic pulling force (Fmech) in motors, etc. For the composition of the axial thrust in a closed impeller that is not in hydrostatic balance (as shown in Figure 2 for the calculation of impeller axial thrust): Here, α is the axial thrust coefficient (based on experience), ρ is the density of the fluid being pumped, g is the gravitational constant (acceleration due to gravity), H is the head, and D2m is the average impeller diameter. The axial thrust coefficient depends primarily on the specific speed (ns; note: this refers to the specific speed used in the EU). For radial and mixed-flow impellers, the following calculation formula applies in the range of 6 rpm < ns < 130 rpm: α = 0.5 × (Dsp/D2m)^3 + 0.09 ≈ 0.1 to 0.3. Here, Dsp is the diameter of the controlled clearance at the suction-side impeller shroud. Figure 2: Mixed-flow unbalanced impeller. This formula is applicable to flow rates (Q) ranging from 0.8·Qopt to 1.0·Qopt, with a gap width of S=0.1 mm. If the gap width is doubled, α increases by about 8 %. Note on pumps: In European countries such as Germany and the UK, the standard unit for specific speed is r/min. The calculation formula is as follows: Here, Qopt represents the flow rate at optimal efficiency, in units of m3/s; Hopt represents the head at optimal efficiency, in units of m; n represents the pump’s rotational speed, in units of r/min; and ns represents the specific speed, also in units of r/min. To convert this into a dimensionless specific speed, the following formula is used: Here, g represents the acceleration due to gravity, which is 9.81 m/s2. For multi-stage pumps equipped with vanes (such as boiler feed pumps), the axial force on the impeller (F1) depends to a large extent on the axial position of the impeller relative to the vanes. In the case of an open radial impeller without a cover on the suction side, the axial force (Fs) is much lower than that of a closed impeller, which means that the axial force of the impeller (F1) is higher. Open impellers with notches in the impeller shroud between adjacent impeller blades generate lower pressure (Fd); therefore, their axial force (F1) is also lower compared to impellers with complete discharge-side shrouds, as shown in Figure 13. For axial propellers (axial impellers, see Figure 14), the axial thrust coefficient (α) is almost equal to the reaction force (rth). The axial thrust can be roughly calculated using the outer diameter (OD) of the axial impeller: The following ratio applies to the axial thrust F1 component of pumps that are geometrically similar at a specified rotational speed (n) and maximum impeller diameter (D2) (see Figure 1: Axial Thrust). In the discharge side and suction side clearances between the impeller and the casing, the rotation of the fluid being processed has a significant impact on the axial pressures (Fd) and (Fs). The average angular velocity of the rotating fluid being processed is approximately half of the impeller speed. Furthermore, due to the Coriolis acceleration (combined centripetal acceleration), the inward flow in the suction side (i.e., external) clearance (side gap) between the impeller and the casing further increases the turbulence in that side gap. In the discharge side (i.e., the internal) clearance of the pump, an outward flow through the clearance occurs due to the lack of hydraulic balance in the impeller (a process opposite to the one described above). The vortex motion slows down, resulting in an increase in the axial force Fd; consequently, F1 increases as well. The axial force on the impeller during startup is higher than that during steady-state operation, because during startup the fluid being processed begins to rotate slowly due to disk friction caused by the braking effect of the impeller shroud or the surface of the stationary casing. Various forms of axial thrust balance: 1) Mechanical balance: The axial thrust is completely absorbed by thrust bearings (such as rolling bearings or tilting pad bearings). 2) Design-based: The impellers are arranged back to back, with thrust bearings used to absorb the remaining axial thrust. 3) Balance or reduce the axial thrust on a single impeller through balance holes, as shown in Figures 7 and 9. 4) Balance the entire rotating assembly using a balancing device with an automatic balancing function (such as a balance disc), or achieve partial balancing through a balance drum and dual balance drums. 5) Reduce the axial thrust on a single impeller by using back blades, as shown in Figure 8. Mechanical axial thrust balancing: The use of rolling bearings to absorb axial thrust is the most effective and economical solution. However, without special balancing equipment, particularly complex thrust bearings must be used; in such cases, the benefits in terms of efficiency and cost may be offset. Design-based axial thrust balance: For example, in a pipeline pump with a 4-stage impeller, there are two sets arranged back to back