Cavitation performance of multistage slurry pump in deep-sea mining

Liquid–gas and liquid–solid phase relationships are established in this study using the theories of cavitation nucleation and solid–liquid two-phase flow, respectively. The relationship between gas and solid phases is then derived, and the effect of solid phase parameter characteristics on the cavitation characteristics of the slurry-conveying slurry in the pump is analyzed. The influence law of particle concentration and speed on the airing performance of two-stage slurry pumps is studied on the basis of computational fluid mechanics. Results show that the cavitation phenomenon reduces the overall pressure of the flow field of deep-sea mining slurry pump. The lowest pressure area is the area of airing development at the entrance of the first-stage impeller blade. The cavitation of the mineral pulp pump becomes evident, and air bubbles rapidly spread over the outlet as the solid phrase particle grows in size. Moreover, solid phase concentration heightens the cavitation of the slurry pump. The cavitation in the pump gradually intensifies as the speed of the slurry pump increases, and a large area of air bubbles sharply forms and disturbs the flow field of the pump when the speed reaches 2000 r/min. In addition, the vortex increases, and the jet phenomenon becomes serious. A comprehensive analysis of the cavitation characteristics of the slurry pump is obtained at the following speed, solid phase volume concentration, and solid phase particle size: n = 1450 r/min, C = 5.3% and d = 20 mm, respectively.

Previous studies on multiphase flow and pump cavitation serve as reference to the present study. This work uses the calculation method of computational fluid mechanics, along with multiphase flow model and dynamic mesh technology. These methods are applied to investigate the principles of the cavitation characteristics of deep-sea mining slurry under multiphase flow conditions and provide a theoretical basis for the structural optimization and parameter selection of deep-sea mining slurry pumps.

Many studies on the cavitation characteristics of various pump types were conducted previously. Pan Q et al. 5 used a full-empty model to obtain the centrifugal pumps’ “head-empty residual” performance curve. Cao Y L 6 numerically analyzed the air-rising characteristics of single-stage axial pumps using filter-based turbulence and Zwart aerification models; they obtained a cavitation characteristic curve that matched their experimental results. Shi W D 7 analyzed the influence of the leaf-top gap of axial current pumps on cavitation characteristics and concluded that the size of the leaf-top gap of the axial current pump increases airing margin. Oddie G et al. 8–12 studied the pressure loss and solid and gas phase distributions in multiphase flow transport and obtained the pump’s external characteristics, flow field pressure, and speed distribution law. Niu X D et al. 13 used the multi-phase lattice Boltzmann model to study the evolution of bubble shape during cavitation. To obtain dynamic characteristics of cavitation and study the relationship between the cavitation and inlet pressure, Wu L et al. 14 used large-eddy simulations (LESs) to calculate the unsteady flow field in the pilot stage with a rectangular nozzle. The above research methods provide an idea for experimentally studying the cavitation performance of slurry pumps.

The performance of slurry pump is affected by several properties, including solid phase particle concentration, particle size, and cavitation characteristics. Moreover, cavitation often disturbs the normal flow of the flow field within the pump, thus causing cavitation damage, vibration, and noise as well as changing the external in situ performance parameters of the pump and even disrupting the entire system in severe cases. 4 Cavitation greatly affects the external characteristics, safety, stability and pump mounting height, and structural design of slurry pumps. Hence, this phenomenon is of great importance to the analysis of the airing performance of deep-sea mining slurry pumps.

Raising ore slurry from the sea floor is one of the core procedures in deep-sea mining. As a key power component of the yang ore system, the slurry pump should be lifted from the sea floor to the surface of the mining vessel. 1–3

The schematic of the deep-sea mining conveyor system is shown in Figure 1 . The pressure required for solid-liquid transport can be supplied by a multistage slurry pump that is raised from the intermediate chamber (point B) to the surface mining vessel (point A).

The rotation of the pump impeller is run by a motor. The circular runner where the hybrid fluid flows through the motor-rotating and extruding in the pump impeller through the combined action of the space-guided leaf transfer and flow-contains substantial pressure energy. This mechanism allows the fluid to flow from the pump outlet to the next section of the tube and then to the sea mining vessel.

The pump consists of two single-stage pumps in series with the motor that are integrated into the outer cylinder body and connected by nuts and bolts. The single-stage pump consists of a space guide leaf and an impeller. Two two-stage pumps in series are installed in a pump body containing a guide shell, and the motor is installed in a motor cylinder with a circular runner. One end of the import and export flanges is connected with the motor barrel, whereas the other end is connected with the Yang mine tube. This scheme serves as the power part of the Yang mine transmission system.

Each stage impeller of the two-stage slurry pump consists of three blades. The 3D solid models of the blade sits and the second-stage pump are shown in Figures 3 and 4 , respectively.

The spatial guide leaves and impellers of the slurry pump are modeled in the Pro-E 3D modeling software and then exported into the ICEM pre-processing software for meshing through the Pro-E interface technology.

The working performance of deep-sea mining slurry pump is characterized by high head, low flow, and large particle size of transported ore. To ensure the overcurrent capacity of the pump runner, the coarse ore that meets the large particle size can pass through the pump runner, and the multistage pump impeller and lead leaf can serve as the amplification flow design in increasing the width of the flow channel in the slurry pump. Moreover, the flow control during the process is low.

where H min is the required head way for the slurry pump (m); ρ w is the freshwater density (kg/m 3 ); P out is the pressure at the outlet of the conveyor tube; and u out is the mixed fluid flow rate at the outlet of the conveyor tube.

As mentioned above, the pressure required for solid–liquid transport can be supplied by a multistage slurry pump. The formula for lifting the hybrid slurry from the intermediate silo to the sea mining vessel is given as

where P 1 and P 2 are the static pressures for sections 1 and 2, respectively (Pa); Z 1 and Z 2 are the vertical heights of sections 1 and 2 (m), respectively; a m1 and a m2 are the kinetic correction coefficients on sections 1 and 2, respectively; Δh m is the energy loss from cross-section 1 to section 2 (J); and u m1 and u m2 are the average flow rates in sections 1 and 2, respectively (m/s).

The sea mining system is complex and involves many influencing factors. The assumptions for the analysis of the mining system are as follows: (1) the base item of the conveying slurry is seawater, and the rest are manganese nodule particles; and (2) the solid–liquid two-phase flow is a continuous uncompressible fluid with constant physical properties, regardless of phase change.

The connection between the gas and the liquid phases is established using the liquid pressure and the bubble radius. In other words, the airing critical point occurs when the liquid pressure and bubble radius are p 2 and r 2 , respectively.

p 2 is the critical pressure at the critical section of slurry pump section 2, whereas r 2 is the critical and the maximum radius of the bubble. The critical state value can be obtained by differentiating formula (11) as follows

The movement of the fluid from position 1 to 2 is an isothermal process, wherein T 1 = T 2 . Given this condition, the above equation (10) becomes

When analyzing the deep-sea mining slurry pump from point 1 to 2, the liquid pressures at locations 1 and 2 are denoted as p 1 and p 2 , respectively.

where N is the gas constant, and T is the absolute temperature (K). With respect to the latter, the new expression for formula (4) is given as

The change in gas pressure p g in the cavitation can be expressed according to either the state equation of the ideal gas or the equation for isothermal process. With the respect to the former, formula (4) can be rewritten as

where p is the liquid pressure (Pa); P v is the saturated vapor pressure of the steam in the cavitation (Pa); P g is the pressure of the gas component in the vacuole (Pa); δ is liquid surface tension coefficient; and r b is the bubble radius (m).

According to the gap wall gas storage mode of Hervey (1947), 16,17 the equilibrated water vapors and other small gases in the isolated spherical vacuoles ignore gas diffusion. Therefore, the bubble static balance condition can be expressed as

According to cavitation theory, the gas core in the liquid will grow explosively when the local pressure of the fluid is lower than the saturated vapor pressure of the liquid at a certain temperature. 15 The high pressure generated when the bubble breaks will affect the structure of the pump body, and continuous action will cause the early fatigue of the pump.

The relationship between the solid and liquid phases is established using the relevant parameters in formula (18) . In conclusion, the solid phase particle concentration and flow volume affect the pressure change in processes 1–2.

To explain the effect of solid phase characteristics on the bubble, the above formula should only contain the solid phase parameters. Substituting formula (16) into (17) and dividing both sides by Q m ρ m g, the formula (17) is given as

When the fluid flows from position 1 to 2 (i.e., the critical cavitation position), the movement can be expressed using the mass-energy equation, that is

where Q m , Q s , and Q w are the slurry, solid, and water flows, respectively (L/s), and ρ m , ρ s , and ρ w are the slurry, solid, and water densities (kg/m 3 ), respectively.

The critical radius of bubble collapse is related to the density of solid phase particles and the flow rate of the solid phase. Keeping other parameters constant, the critical radius r 2 of bubble collapse decreases as the solid particle density ρ s and the flow rate Q S increase.

With pressure as intermediate amount, the relationship between the solid and the gas phases can be established by transforming the parameters. Substituting formula (15) into (18) gives

Only the first and third phases will undergo mass exchange when the pressure is lower than the cavitation pressure. Moreover, phase change takes place.

The first, second, and third phases involve seawater, manganese nodule particles, and air, respectively. The nodule particles and bubbles are uniform spherical particles considering the drag force among the three phases and the velocity slips among the phases.

The fluid in the solid–liquid two-phase flow is a continuous incompressible fluid, and the physical properties of the slurry medium are constant. Moreover, no temperature difference exists in the flow field, and no heat exchange occurs.

Given the actual working conditions of the deep-sea mining system operation and the feasibility of the numerical calculation of the cavitation’s steady state as well as the accuracy of the results, the following assumptions are made:

As can be seen from Figure 6 , with the increase of the number of model meshes, the solid phase volume concentration of each monitoring point shows an increasing trend. When the number of meshes increased from 88.9 million to 267.8 million, the solid phase volume concentration of three monitoring points increased greatly. When the number of meshes increased from 267.8 million to 352.5 million, the increase of solid phase volume concentration was smaller in three monitoring points. Considering the calculation accuracy and the calculation time cost, the paper selects the mesh type with a mesh number of 267.8 million for numerical simulation.

In order to verify the influence of mesh size on the calculation results, four kinds of sizes were used to divide the model, and the corresponding mesh number was A (88.9 million), B (155.2 million), C (267.8 million) and D (352.5 million). Using four kinds of meshes to simulate the same working condition, analyze the law of solid phase concentration varies with the number of grids. The specific coordinates of three points are M1 (slurry pump inlet center), M2 (slurry pump first stage impeller center) and M3 (slurry pump outlet center). As shown in Figure 6 , the relationship between the solid phase volume concentration and the number of meshes at three monitoring points is obtained.

A section of the inlet and outlet water pipes is added at the inlet of the impeller runner and the outlet of the space vane runner to allow the development of fluid and consequently improve the accuracy of numerical calculation. Data transmission was performed among the respective watersheds, between the inlet pipe and impeller flow path inlet, between the impeller flow path outlet and the space guide vane flow path inlet, and between the space guide vane flow path outlet and the external parts. This process was performed by establishing an interface between the water pipes to achieve dynamic and static couplings between the static and the moving meshes to consider the rotating feature of the impeller. The calculation model after meshing is shown in Figure 5 .

This study utilized the ICEM preprocessing software for mesh division. First, a topological rectangular block is created for a relatively regular part of the single flow channel. The topology block is then extended according to the actual shape of the model, and the excess block is cut off. After creating the block, the points, lines, and faces between adjacent blocks are duplicated to obtain a block structure similar to the actual one. Finally, the block is used to generate structure mesh. The number of nodes and meshes, node density, and the near-wall mesh spacing are locally encrypted. In addition, the spatial vanes are meshed.

The steady-state numerical simulation is performed using the Euler–Eulerian model, which is based on the time-averaged Navier–Stokes equation and can calculate arbitrary particles and continuous phase materials. Slurries behave as typical turbulent flows during the transportation process. The fluid under turbulent flow not only obeys the momentum, energy, and mass conservation, but it also satisfies the turbulence equation. The multistage pump, which delivers slurry without considering heat exchange, mainly follows the momentum, continuity, and turbulence equations.

where i and j are coordinate directions; u i is the liquid phase velocity vector (m/s); ρ s is the solid phase density (kg/m 3 ); and P is the equivalent pressure that considers centrifugal force (Pa).

Mass exchange occurs between the gas and liquid phases during cavitation. The conversion between the phases is governed by the following equations:

where v represents gas phase; α is the gas phase volume fraction; R e and R c are mass transfer sources that are respectively related to the growth and collapse of vapor bubbles and describe the mass transfer between the liquid and gas phases.

where RB is the bubble radius (m); δ is the liquid surface tension coefficient (N); ρl is the liquid density (kg/m 3 ); P v is the bubble surface pressure (Pa); and P is the local far field pressure (Pa).

Deep-sea mining slurry pumps transport seawater slurries containing manganese nodules. The physical properties of seawater are related to geographical distribution. The Chlor-Alkali Industry Physical and Chemical Constant Manual and the Handbook of Thermophysical Properties of Commonly Used Materials defined an average salt concentration of 3.5% as the physical properties of a 3.5% sodium chloride solution in seawater. The physical properties of each phase are listed in Table I .

The simulation condition are as follows: working flow Q = 420 m 3 /h, solid phase concentration = 8%, rotating speed = 1450 r/min. Using the equilibration method, 18 different particle size conditions with sizes between 5 and 36 mm are obtained.

Figure 8 shows the isostatic contour of the maximum axial section of the impeller. The pressure gradient on the section uniformly increases from a minimum pressure of −200,000 to 400,000 Pa along the direction of the streamline. With the increase in particle diameter, the low-pressure area change exhibits the same regularity as the blade pressure. Based on the pressure distribution, the distribution of the gas phase should be at the inlet of the impeller blade’s back. The distribution of the gas phase is consistent with the law of static pressure distribution as the particle size of the solid phase changes.

The low-pressure area gradually expands to the outlet at the back of the blade as the particle size increases from 10 to 25 mm. Large particle size signifies great inertia, velocity difference in the water flow, and low-pressure area in the vicinity of the solid phase. However, the low-pressure area decreases as the particle size increases because the particle size is larger under the same concentration per unit volume.

Figure 7 shows the static pressure distribution of the first stage impeller under different particle sizes. The static pressure distribution of the positive rear of the blade increases along the direction of the flow line. The lowest pressure point appears at the entrance of the blade’s rear because under the action of centrifugal force. The velocity of the fluid at the inlet rapidly increases and smashes against the blade, forming a low-pressure region that facilitates the occurrence of vacuoles.

Pressure is necessary to induce cavitation. The pressure field in the first stage impeller basin is analyzed. In this study, the working conditions of particle sizes 10, 15, 20, 25, 30, and 35 mm are selected for specific analyses due to the large amount of data and limited space. In addition, the maximum section of the impeller is intercepted for process analysis.

Figure 9 shows that the gas phase is mainly concentrated at the inlet of the blade’s rear. Thus, the occurrence of cavitation is mainly caused by the pressure drop. The gas phase distribution range is small when the particle diameter is 10 mm. Moreover, the volume fraction is below 0.3, and the gas phase distribution is relatively dispersed. As the particle diameter increases to 25 mm, the gas phase distribution area rapidly expands to about half. In addition, the particle size exerts a significant effect on cavitation, and the maximum gas phase volume fraction reaches 0.9. If the gas phase region is connected into a sheet, then the high concentration of the bubbles would disturb the internal flow field of the slurry pump and would greatly damage the blade. Moreover, the diameter continues to increase. In addition, the gas phase distribution area slightly decreases, but the volume fraction remains high.

NPSHa gradually decreases as particle size increases. Hence, the cavitation and energy loss increase, whereas the lift decreases. Moreover, the minimum pump’s cavitation margin is observed (1.11 m) when particle size equals 25 mm. The cavitation at this time is the most intense, and the head achieves the smallest value (56.14 m). The solid phase particles then continue to increase, and the anti-cavitation characteristics are improved and maintained at about 3.5 to 4.5 m. The number of particles per unit volume decreases as the particle size expands at the same volume concentration, thereby degrading the effect on the gas phase.

The numerical simulation of the inlet and outlet pressure values of the slurry pump under different particle sizes is combined with the cavitation allowance and the theoretical formula of the lift to calculate the pumping cavitation allowance NPSHa and the lift H ( Table II ). The deep-sea mining slurry pump has the best anti-cavitation characteristics of 8.16 m and a maximum head of 70.3 m when the particle size is 5 mm.

This section presents the numerical simulation of the full solid–liquid two-phase flow with particle volume concentrations of 4%, 6%, 8%, and 10% under the following conditions: working flow Q = 420 m 3 /h, particle size = 20 mm, and rotating speed = 1450 r/min.

The maximum axial section of the selected impeller is processed, and the static pressure contour map is shown in Figure 11 . The lowest pressure (approximately −150,000 Pa) is distributed at the back of the blade, whereas the highest pressure (approximately 350,000 Pa) is distributed at the front exit of the blade. Furthermore, the low-pressure area consistently changes with the leaves when the concentration of the nodule particles is increased from 4% to 10%.

When the concentration of the nodule particles is 4%, only a small portion of the low-pressure zone exists at the entrance in the blade’s rear. The low-pressure region at the inlet of the back side of the blade increases when the concentration is increased to 10%. By contrast, the low-pressure region expands as the solid phase concentration increases. When the water flow velocity in the pump decreases, the velocity of the particles will be greater than the water flow under the action of inertia. In addition, a speed difference is formed between the two, thereby decreasing the pressure of a portion of the solid phase particles. The solid phase number per unit volume heightens with particle concentration. Moreover, the low-pressure area will widen, providing favorable conditions for the development of cavitation.

Figure 10 shows the static pressure distribution of the first-stage impeller blades of the deep-sea mining slurry pumps under different solid phase concentrations. The pressure gradually increases along the flow direction of the suction and pressure surfaces of the blade. The lowest pressure point appears at the entrance of the back of the blade, forming a low-pressure zone that is prone to cavitation.

At a nodule particle concentration of 4%, the gas phase volume fraction is less than 0.1, but the distribution range is wide. The gas phase volume fraction rapidly increases with particle concentration. When the concentration increases to 10%, the low gas phase volume fraction region at the back entrance of the blade reaches 0.9, which causes severe cavitation to the impeller. Moreover, the distribution area of the bubbles is not continuous like the low-pressure zone. The distribution of the gas phase becomes increasingly dispersed as the particle concentration increases. The increase in solid phase particles inhibits the appearance of additional bubbles.

Figure 12 illustrates the gas phase volume distribution of the first stage impeller during the cavitations under different particle concentrations. The gas phase is mainly distributed in the low-pressure zone at the back inlet of the impeller blade. The bubbles gradually disappear along the direction of the flow line, which is consistent with the trend of pressure change and indicates that pressure reduction is necessary for cavitation.

When the concentration of the solid particles is 4%, the head of the slurry pump and cavitation allowance are 61 and 9.2 m, respectively. At this time, the slurry pump demonstrates a good anti-cavitation performance. The severity of energy loss increases with the concentration of the solid phase particles due to the abrasive action of the particles on the pump and the friction between the particles. Moreover, the cavitation damage causes the lift to gradually decrease. When the solid phase concentration reaches 10%, the lift decreases to 50 m, and the cavitation margin drops to a minimum of 0.8 m. Both parameters behave linearly with the change in concentration.

The internal cavitation and working characteristics of the slurry pump at high and low speeds are analyzed under the following conditions: particle size = 20 mm, solid phase volume concentration = 8%, and flow rate Q = 420 m 3 /h. Other parameters are kept constant, and the full-flow cavitation numerical simulation of the solid–liquid two-phase flow field is performed at rotational speeds of 960, 1450, and 2000 r/min.

On the one hand, the pressure drop at the suction surface reaches a minimum value of −248 kPa as the rotational speed increases to 1450 r/min. On the other hand, the lowest pressure at the suction surface is basically equal to the pressure at 1450 r/min as the rotational speed value reaches 2000 r/min. However, the increase in the length of the flow line increases the pressure. In conclusion, at 2000 r/min, a large area of the suction side of the blade belongs to the low-pressure region, consequently widening the bubble distribution area.

The pressure of the mixed fluid on the pressure side of the impeller blade sharply increases with the rotational speed. The fluid pressure rise on the pressure surface increases from 120 kPa at 960 r/min to 458 kPa at 1450 r/min. As the speed reaches 2000 r/min, the pressure at the pressure surface reaches a maximum value of 460.4 kPa. In conclusion, the effect of the kinetic energy of the impeller on pressure energy becomes obvious as rotational speed increases. At a speed of 960r/min, the total pressure of the slurry on the suction surface of the blade increases with the length of the flow line and exceeds 300 kPa. From the perspective of the pressure, the rotation speed is not influenced by cavitation.

The static pressure data of the intermediate flow line of the suction and pressure surfaces of the first stage impeller are extracted and plotted ( Figure 14 ). The fluid pressure on the suction side of the impeller blade is lower than that on the pressure surface, and the rotational speed increases from 960 to 1450 r/min. The pressure difference between the two sides increased from 12 to 221.8 kPa, and the pressure difference between the two sides reaches 347.3 kPa at a rotational speed of 2000 r/min.

Figure 15 depicts the gas phase volume fraction of the first stage impeller blades under different speeds. Cavitation mainly occurs at the back entrance of the impeller blade. No cavitation is present when the rotational speed is 960 r/min, which is consistent with the predicted outcome of the pressure analysis. When the rotation speed is increased from 1450 to 2000 r/min, bubbles start to appear at the entrance of the suction side of the blade. By contrast, the bubbles along the direction of the flow line gradually disappear.

A large area of air bubbles appears and occupies about 3/4 of the blade when the rotational speed is 2000 r/min. The gas phase volume fraction at this time is around 0.9. The appearance of the bubbles can be attributed to the fluid delivered by the slurry pump, which is governed by the solid–liquid two-phase flow. In addition, the manganese nodules particles are not infiltrated and exhibit a larger tensile stress than the fracture of the seawater medium under the action of virtual mass force and high rotational speed. The fluid has high kinetic energy, which causes the partial pressure drop and promotes cavitation. Moreover, the particles demonstrate great inertial force at high rotational speeds. When the fluid velocity in the impeller is slightly reduced, the liquid and solid phases will differ in density. This effect induces velocity difference between the phases and reduces the pressure near the particles, thus promoting cavitation. High speed signifies high speed discrepancy between the two phases as well as obvious effect.

The maximum axial section of the impeller is then selected, and the velocity streamline is analyzed. Figure 16 shows that the maximum speed appears at the inlet of the suction side of the blade, and the velocity gradually decreases along the direction of the streamline. However, a high-speed region is formed at the exit of the pressure surface adjacent to the blade with a cross-section that resembles a jet-wake structure.

When the rotational speed is 960 r/min, the maximum speed is 12 m/s. The flow line in the whole area is even and stable, indicating that the fluid flow is stable at this speed. When the speed is increased to 1450 r/min, the maximum speed increased by 9 m/s. The cavitation causes a vortex in the flow field, thereby increasing the field’s instability. When the speed reaches 2000 r/min, the maximum speed is as high as 40.2 m/s. Consequently, the vortex increased and a serious jet region is formed in the flow passage. The area exhibits high speed and low pressure, and the cavitation phenomenon is intense. The high-speed bubbles simultaneously induce severe cavitation damage to the blades and vibration or surge on the entire unit.