Figure 1 shows the structural system of the BDDCPS. The BDDCPS is composed of two parts: a bearingless permanent magnet slice motor and a centrifugal pump with an “L”-shaped vertical structure. There are two sets of windings on the stator, namely torque winding and suspension force winding. They are used to generate a suspension bias magnetic field. The rotor permanent magnet and centrifugal pump impeller are integrated and placed together in the rear chamber of the centrifugal pump, which is driven by the motor magnetic field for operation. A displacement sensor is installed in the air gap of the motor to detect the positional information of the impeller rotor. When the BDDCPS operates normally, the impeller rotor is in a magnetic levitation high-speed rotational state, which can realize shaftless rotation. The torque winding part of the BPMSM provides the power for pump rotation, and its torque winding part is equivalent to a regular permanent magnet synchronous motor. Therefore, the motion equation of BPMSM can be represented through the application of a permanent magnet synchronous motor, as follows:
where
Te is electromagnetic torque;
TL is load torque, which is the pump;
J represents the moment of inertia;
ωr represents rotor angular velocity; and
pM is the torque winding’s pole pairs.
2.1. Axial Suspension Principle
When the permanent magnet thin rotor of the motor is flush with the center position of the stator teeth, the magnetic circuit in the air gap is shortest, and the magnetic resistance is at a minimum. When the permanent magnet rotor deviates or twists from the balance position due to external force interference, the air gap balance is broken, and the magnetic resistance increases. According to the “magnetic resistance minimum principle”, in order to minimize the magnetic resistance of the magnetic circuit, the magnetic field will generate a magnetic pulling force in the opposite direction of the external force, forcing the rotor to return to its equilibrium position.
As shown in
Figure 2a, when the motor rotor deviates up and down at the equilibrium position, the electromagnetic force in the x direction after orthogonal decomposition is zero, the resultant force in the y direction is
y0, and the magnetic force
y0 pulls the rotor back to the equilibrium position.
When the motor rotor occurs in the axial torsion, as shown in
Figure 2b, the average air gap is changed; then, the electromagnetic force produces a torque in the opposite direction of motion to realize the return of the rotor. To sum up, it can be seen that the rotor can achieve passive suspension in the axial direction of 3 degrees of freedom based on the “magnetic resistance minimum principle”.
2.2. Radial Suspension Principle
Figure 3 shows the principle of radial suspension force production under the no-load condition. The motor combined 4-pole torque windings
NMa,
NMb and 2-pole suspension windings
NBa,
NBb in the same stator slots. The currents in the 4-pole torque windings generate 4-pole magnetic field
ΨM, and the currents in the 2-pole suspension windings generate 2-pole magnetic fluxes
ΨB. When the 4-pole magnetic field generated by torque windings and the 2-pole magnetic field generated by suspension windings interact with each other, the flux’s density in the negative direction of the x-axis increases, but the flux’s density in the direction of the x-axis decreases. Therefore, the unbalanced gap flux density results in the production of the radial force
Fx in the negative x-direction. In addition, another 2-pole suspension winding, which is perpendicular to windings, can produce the radial force in the y-direction. Therefore, the size and direction of the radial suspension force can be controlled by controlling the size and direction of the current of the suspension windings, and the stable operation of the motor can also be realized [
16,
17].
2.3. Three-Dimensional Model of the BDDCPS
In the BDDCPS, the impeller rotor mainly relies on the magnetic field of BPMSM for suspension. After the above verification, the BPMSM can control the movement of the impeller reliably and effectively. This is the usual situation when the system runs a stable operation after the start-up process. BDDCPS is a multivariable, nonlinear, and strongly coupled system. Realizing the dynamic equilibrium of the impeller rotor not only simplifies the difficulties of calculation and decoupling but is also the key to ensuring smooth operation of the system. Compared to stable operation, parameters such as flow rate, head, pressure field distribution and velocity field distribution will change dramatically in a short time during the start-up process. Because the BDDCPS adopts a special structure of magnetic suspension, the strong and dramatic changes mentioned above can cause excessive hydraulic impact, leading to significant displacement of the impeller rotor [
18,
19]. It is a fatal problem that can cause a severe collision between the impeller and the pump casing, unnecessary impeller wear, deteriorating system performance, and reductions in service life [
20]. In particularly severe cases, there is a risk of device suspension failure, leading to system collapse. This is catastrophic for the semiconductor manufacturing industry, which aims for precision, and for the biomedical field, where safety comes first. Only by addressing these issues can BDDCPS make progress in the aforementioned fields. Therefore, it is crucial to maintain smooth changes in the BDDCPS during the start-up process within a short time. In other words, uniform changes in external characteristics, pressure fields and velocity fields are crucial during the start-up process. In order to ensure that the centrifugal pump can smoothly start the pump, this paper set three different start-up methods to discover the optimum operation of the pump start-up method.
The characteristic parameters of the pump are shown in
Table 1. The model pump is a single-stage single-suction centrifugal pump, with clean water as the flow medium and a density of
ρ = 1000 kg/m
3.
By referring to the relevant empirical design formulas in the “Modern Pump Theory and Design” [
21], the reference external dimensions of each component of the pump can be calculated. The parameters are shown in
Table 2.
According to the structural parameters described in
Table 2, a three-dimensional model of the flow-field calculation domain of the centrifugal pump is constructed using the mechanical design software, UG 12.0. The main components include the impeller, volute, pump chamber, inlet and outlet, and its extension section. Considering that the permanent magnet rotor and centrifugal pump impeller of BPMSM are integrated structures, the rear chamber of the centrifugal pump has been redesigned. The rear chamber is slightly longer than the axial length of the permanent magnet rotor to facilitate the placement of the permanent magnet rotor. The model pump is equipped with an inlet extension section of 6 times the pipe diameter and an outlet extension section of 8 times the pipe diameter. The three-dimensional model of the entire centrifugal pump flow field is shown in
Figure 4. The radial eccentric displacement of the rotor is shown in
Figure 5. The black circle indicates that the rotor is in the center position. The red circles show the position of the rotor at the maximum displacement of the
X-axis. The maximum radial eccentric displacement is 3 mm, the diameter of the spiral casing base circle is 78 mm, and the outlet diameter of the impeller is 70 mm. This means that the minimum gap between the impeller and pump head is 1 mm.
2.4. Numerical Calculation
Due to the non-uniform spatial distribution of the internal flow field of the pump, the turbulent and irregular flow characteristics may make it difficult to directly solve the N-S equation (the most basic fluid motion equation). Therefore, the turbulence model is introduced to study this issue.
According to the number of differential equations, turbulence models can be divided into two equation models, one equation model, and zero equation models. Among them, the series k-ε (standard k-ε, RNG k-ε, and realizable k-ε) and series k-ω (standard k-ω and SST k-ω) are widely used in the numerical calculation of fluid machinery.
Considering that the internal flow of the pump is three-dimensional and involves high Reynolds number turbulence, this study needs a stable and widely applicable turbulence model. The standard
k-ε model can predict complex fluid length scale distributions and solve the turbulent stresses through the two equations of turbulent kinetic energy and dissipation rate [
22]. At the same time, the standard
k-ε is widely used for solving engineering flow field problems because of its good efficiency and economy. Therefore, the standard
k-ε model is selected to simulate the transient start-up process of the pump.
Assuming that the flow inside the pump is the fully developed turbulent flow, ignoring the molecular viscosity characteristics, the turbulent viscosity
μt is expressed as follows:
The turbulent kinetic energy
k and dissipation rate
ε can be expressed, respectively, as follows:
where
Gk is the turbulent kinetic energy generated by the average velocity gradient;
Gb is the turbulent kinetic energy generated by buoyancy;
Ym is the effect of turbulent pulsation expansion on the total dissipation rate;
uj is the velocity of flow;
C1ε,
C2ε,
C3ε are 1.44, 1.92, and 0.09, respectively;
Cμ,
σk and
σε are 0.09, 1.0 and 1.3, respectively [
23].
The grid is the foundation of discretization of the control equation space, and grid independence verification is an effective way to test whether the number of grids affects the calculation results of the model. The ICEM-CFD module in ANSYS 2021R2 software was used to complete the grid division of each calculation domain of the model pump. Considering the significant degree of distortion in the blade structure and the complexity of the structure at the volute tongue, tetrahedral unstructured grids are selected for the two main flow components of the model pump, as shown in
Figure 6. The mesh refinement is performed on the tongue area of the volute.
To verify the independence of the grid, this paper has established 5 sets of grid models with different numbers and has analyzed the calculation results of head and efficiency, as shown in
Table 3. When the total number of grids is over 6.34 million, there is no significant difference in the efficiency calculation results. The amplitude of head change tends to stabilize with the increase in grid numbers; here, the amplitude of head change is less than 2%. The impact of grid differences on the calculation results can be ignored. Considering the limited computing resources, the total number of grids selected for the model pump in this study is 6.34 million.
ANSYS CFX 2021R2 software was used to conduct unsteady numerical simulations of the transient start-up process of the model pump. The pressure inlet p is set to 1 atm, and the mass flow outlet ms to 3.8888 kg/s. The surfaces of the blade and the front of the impeller, as well as the surfaces of the blade and the rear cover plates of the impeller, are designated as rotating wall surfaces. The wall surfaces of the stationary components are assigned non-sliding wall conditions. The dynamic and static interfaces are set at the intersection of the rotating and stationary domains, specifically between the impeller and the volute, the impeller and the rear chamber, and the impeller and the inlet section. The frozen rotor connection mode is utilized. For its wide applicability and high stability, the standard k-ε turbulence model is chosen for subsequent research. In the ‘Global Initialization’ module of the ANSYS CFX software, the initial speed and initial time were set to zero to define the initialization conditions. The residual convergence accuracy of the simulation is 10−4, with a time step of 4° rotation (i.e., 0.0001111 s). It simulates 22 rotation cycles, resulting in a total duration of 0.2199978 s.