Influence of wall roughness on pressure distribution and performance of centrifugal dredge pumps

Wall roughness is an important factor that affects pump head and efficiency. Based on the open source program OpenFOAM, the wall function method and the equivalent sand roughness model were used to study the influence of wall roughness on the performance of centrifugal dredge pumps. Then, the steady state of the dredge pump under 10 different types of wall roughness was calculated using the k-omega SST turbulence model. The pressure distribution under different flow rates and rotation speeds was also analyzed. The results reveal that the relationship between the increase of wall roughness and the performance of the dredge pump can be roughly divided into three stages, which may be related to the link between roughness and the viscous layer. In particular, the pump performance decreases significantly, and the shaft power increases linearly when it is in the third stage (complete rough region). Finally, different degrees of roughness have minimal effects on the pressure distribution of the impeller.


Introduction
Large centrifugal dredge pumps have been widely used as an essential device in the dredging industry and in transporting resources, such as ore and coal.With the rapid development of these fields, the requirements for dredge pumps have become increasingly higher in recent years (Wen et al. 2019).However, many studies (Cavazzini et al. 2015;Cao et al. 2018) on the performance prediction of dredge pumps have not mentioned or considered the influence of wall roughness.The surfaces of the impeller and volute have certain degrees of roughness required in engineering practice (Yamazaki et al. 2006;Yun and Kim 2014), and the erosion behavior inside the centrifugal dredge pump can also lead to an increase in wall roughness.Therefore it is critical to determine the appropriate surface treatment of the model pump in experiments and numerical research.
Similar to the flow in the pipeline, the energy loss in a pump depends on the Reynolds number and relative wall roughness, which means that the pump's efficiency is affected by pump size, fluid velocity, fluid viscosity, and wall roughness (He et al. 2019).Therefore, wall roughness is increasingly vital in improving pump efficiency (He et al. 2019;Wang et al. 2021).Stepanoff (1980) stated that if the volute channel is cleaned, efficiency can be improved by 2%-4% for small centrifugal pumps.Lomakin (1978) and Aldaş and Yapici (2014) found that efficiency can be significantly improved when the wall roughness is considered.However, Li and Wang (2014) conducted a numerical study on the influence of wall roughness of an axial flow pump and found that pump efficiency decreased from 84% to 70% when the wall roughness increased from 0 to 0.6 mm.Meanwhile, Lim and Sohn (2018) studied the influence of impeller wall roughness on centrifugal pump efficiency through experiments and found that impeller blades are susceptible to roughness.To identify the reasons for the differences in research results, this study investigated the roughness and performance of a mud pump based on a large-scale mud pump.
In addition, some researchers (Gülich 2003;Xu et al. 2018;Li et al. 2020) have examined the influence of roughness on the pressure and flow field in the pump.For example, Deshmukh and Samad (2019) analyzed the influence of different roughness levels on the turbulent energy and eddy viscosity characteristics of electric submersible pump, thereby demonstrating its influence on pressure distribution and velocity field.Zhu et al. (2006) studied the influence of wall roughness on the flow field characteristics of axial flow pumps and performed simulation calculations to obtain the performance differences of axial flow pumps under the influence of different degrees of wall roughness.
In summary, wall roughness has a specific influence on the performance of a model pump.Therefore, from the perspective of the large centrifugal dredge pump and pressure field, the current paper analyzes the influence of wall roughness on pump performance and pressure field in the pump, mainly in terms of different flow rates, rotation speed, and wall roughness.Our findings can provide more accurate numerical references for future performance research on centrifugal dredge pumps.

Numerical model and validation
In this paper, we investigated the steady flow field structure and pressure distribution of centrifugal dredge pumps with different levels of roughness based on the open-source program OpenFOAM.We applied the SST k-omega turbulence model during the steady numerical simulations of the dredge pump to solve the pressure field of the dredge pump.

Flow equations and turbulence model
The flow governing equations are given by: where x i and x j are the Cartesian coordinate components, and p, u i , t, and S ij represent pressure, velocity, time and the average stress tensor, respectively.Furthermore, we used the SST (Shear-Stress Transport) k-ω turbulence model to close the N-S equations.Details of the turbulence model can be found in Gritskevich et al. (2012) and Zhang et al. (2020).

Physics model and mesh
The three-blade dredge pump selected in this paper is mainly divided into six parts: inlet, hub, shroud, blades, volute, and outlet.In order to ensure the accuracy of the dredge pump's performance, the impeller clearance is considered.The main design parameters of the dredge pump are shown in Table 1.The dredge pump is a closed impeller with three-dimensional twisted blades.The design flow rate Q d was 3.5 m 3 /s, and the nominal head H d at the design flow rate was 60 m.In addition, and the nominal rotation speed n d was 270 rpm.The general view of the dredge pump is shown in Fig. 1.
The structured grid generated by the Pointwise software for the dredge pump was adopted to precisely capture the complex flow patterns of the dredge pump.Figure 2a shows the topological structure of the dredge pump grid, while Fig. 2b is an enlarged view of a part of the impeller grid.The near-wall treatment of the mesh directly influences the final numerical results, and the impeller grid of the dredge pump is refined.Figure 2a shows the grid topology of 1/3 the impeller.The height of the first grid near the wall was set to about 0.001 m (Fig. 2b), and the ratio of the grid length of the dredge pump was less than 3. Finally, about 35 million grids were generated for the impeller, and the total grid number of the dredge pump was about 65 million.After calculation, the average y + value of the impeller was about 70.In addition, the tool check-Mesh was used to indicate the most important quality characteristics and the primary types of cells.

Roughness model
In dredge pump products, the roughness varies in shape and size.To solve this problem, OpenFOAM used equivalent sand-grain roughness (K s , 0 for smooth walls) to describe the roughness characteristics of the metal surface.As shown in Fig. 3, a rough wall can be equivalent to a sphere with an average height of K s , which is densely distributed on a smooth surface.
Based on turbulence kinetic energy's theory, when the rough wall function is used for the rough wall in the numerical model, the boundary conditions of the rough wall function constrain the wall with turbulent viscosity.The boundary condition controls the roughness effect through the wall roughness parameters.
The model expression is expressed as: (3) where k is the turbulent kinetic energy (m 2 /s 2 ).y is the wall-normal height (m).y + is the estimated wall-normal height of the cell center in wall units.
C μ is the empirical model constant [-].ν w is the kinematic viscosity of fluid near the wall (m 2 /s).ν tw is the turbulent viscosity near the wall (m 2 /s).κ is the von Kármán constant [-].E is the wall roughness parameter [-].
Eis the modified wall roughness parameter [-].K s is the sand grain roughness height.K s + is the sand grain roughness height in wall units.f n is the roughness function parameter.C s is the roughness constant [-].u * is the shear velocity (m/s).K s represents the average height of the uniformly laid sand grains in the equivalent sand grain model, which differs from the actual wall roughness (Ra).Adams et al. (2012) proposed an algorithm to convert the measured wall roughness parameters to equivalent sand grain roughness, used it to convert the wall roughness to equivalent sand grain roughness, and subsequently performed experiments to verify the model.Following their research, they determined the relationship as follows: In this study, 10 types of roughness are defined based on the actual wall roughness Ra of the metal 3D printing and casting process.Table 2 presents the specific conversion between Ra and K s , flow rate Q, and rotating speed n cases of this study.

Numerical verification
To verify the numerical simulation method, the head and efficiency were verified based on the performance curve in the dredge pump manual.As shown in Fig. 4a and b, under the conditions of rotating speed n of 210 and 230 rpm, there is a small difference between the head and efficiency of the dredge pump.Furthermore, the maximum difference is about 2.7% at Q = 3.0 m 3 /s.When the rotating speed n is 270 rpm, a huge gap is observed between the numerical model results and the (4) K s = 5.863Ra performance curve in the manual, with the biggest gap of about 5% found at Q = 2.0 m 3 /s.In summary, the numerical model can effectively simulate and predict the performance characteristics of the dredge pump.

Results and discussion
The rotating speed must be frequently adjusted during the actual operation of the dredge pump to optimize the efficiency of the whole dredging work.Figure 5 shows the pressure field distribution on the impeller's middle slice at different speeds under the Q = 3.0 m 3 /s.As shown in the figure, the negative pressure field inside the impeller gradually expands and reaches a stable level at 270 rpm with the continuous improvement of the rotating speed.In addition, the stabilized pressure field is not symmetrical due to the asymmetric structure of the volute, although the negative pressure area at the outlet of the volute is large.Meanwhile, Fig. 6 shows the pressure distribution on the blade and hub in the impeller, in which there is increased pressure from the blade's leading edge to the tip, especially at the latter.Furthermore, the pressure around the blade tip in the flow passage is higher than in the other hub positions.
Flow rate fluctuation often occurs in the actual operation of the dredge pump.Figure 7 shows the distribution of the pressure field on the middle slice of the dredge pump under different flow rates at n = 310 rpm.As shown in the figure, the negative pressure field area gradually decreases with the increase in flow rate.Furthermore, the internal pressure field gradually tends to be symmetrical with the flow rate of 4.0 m 3 /s as the boundary.Similarly, the pressure distribution on the blade and hub in Fig. 8 shows that the pressure surface of the blade is larger than that of the suction surface in terms of area and size.
The actual metal processing of the dredge pump often involves the treatment of the wall roughness of the dredge pump.In accordance with the roughness involved in different processes of metal 3D printing and casting products, the distribution of the dredge pump's internal pressure field under different levels of wall roughness was studied at n = 310 rpm and Q = 3.0 m 3 /s.Upon comparing the pressure fields with different roughness levels in Figs. 9 and 10, we find that the surface with different roughness levels has little influence on the large dredge pump.Furthermore, the shape of the pressure distribution is slightly different, only at the blade's tip.
Figure 11 shows the efficiency change of the dredge pump under different roughness levels.By analyzing the broken line of 2.5 m 3 /s flow rate in the diagram and combining the theory of roughness and viscous layer, we find that efficiency can be divided into three stages with the increase of roughness.First, in the hydraulic smooth region (K s < 8), efficiency changes minimally when the efficiency is within a small roughness range.This may be attributed to the low level of roughness, which has little influence on the pressure and flow field in the pump.Second, in the transitional rough region (8 < K s < 50), a fluctuation in efficiency is observed (1.5%).This may be due to the fact that the roughness level is higher than that of the boundary layer (transitional roughness region), which affects the pump's pressure and flow pattern stability.Finally, efficiency suddenly drops at K s > 50, which is due to the roughness being too high, thereby hindering water flow (hydraulic rough region).In addition, the first stage's roughness boundary gradually decreases as the flow rate increases.
Figure 12 shows the head change of the dredge pump under different roughness levels.As can be seen from Fig. 12, the influence of roughness on the head is also divided into three stages.First, roughness has little influence on the lift in the hydraulic smooth region (K s < 8).Second, the lift fluctuates (3%) as the roughness level continues to increase in the transitional rough region (8 < K s < 50).In a complete rough region (K s > 50), the head tends to be stable but smaller than that of the dredge pump under the smooth wall.
The shaft power P can reflect the change in the cost of the dredge pump during operation; thus, the change of shaft power P with roughness is given.Shaft power can be calculated using Eq. ( 5).As shown in Fig. 13, the change of shaft power is roughly divided into two stages.In the first stage (the hydraulic smooth region and transitional rough region, K s < 50), the shaft power changes minimally with the increase of roughness.This finding shows that roughness has little influence on the shaft power of the dredge pump in the hydraulic smooth zone and the transition zone.In the second stage (the complete rough region, K s > 50), the shaft power increases linearly with the increase of roughness.This finding indicates that when the roughness increases to a certain extent and hinders the movement of water flow, it will greatly impact the shaft power: where M i is the average axial torque of the impeller.

Conclusions
A numerical study was conducted in this work to investigate the influence of wall roughness on the centrifugal dredge pump performance and pressure distribution.The characteristics of dredge pump performance with different wall roughness levels were also studied.The internal relationships among operating performance, pressure field, and wall roughness are also identified.The study's conclusions can be drawn as follows: (1) The impeller's negative pressure field gradually expands, reaching a stable level at 270 rpm, with the continuous improvement of rotating speed.
With the increase in flow rate, the area of the negative pressure field decreases gradually.With the flow rate of 4.0 m 3 /s as the boundary, the internal pressure field gradually tends to be symmetrical.The pressure field with different roughness levels shows that the surfaces with different degrees of roughness have minimal influence on the large dredge pump.Furthermore, the shape of the pressure distribution is slightly different, only at the blade's tip.(2) According to the theory of roughness and viscous layer, efficiency and head can be divided into three stages (K s = 8, 50 μm) with the roughness increase.
In addition, the variation of shaft power with roughness can be roughly divided into two stages (Ks = 50 μm).Finally, the relationship between roughness height and boundary layer height mainly causes the changes in different stages.
(5) P = M i × n The results further indicate that different wall roughness levels affect the performance of the dredge pump.In future studies, the dredge pump's internal flow field will be examined based on this study's results, and the blades of the dredge pump will be optimized to achieve a more efficient and stable operation.

Fig. 1 Fig. 2 Fig. 3
Fig. 1 General view of the dredge pump.a stereo view; b sectional view

Fig. 4 Fig. 6 Fig. 7 Fig. 8 Fig. 9
Fig. 4 Efficiency and head comparison between the numerical and experimental results of the pump.a efficiency; b Head

Fig. 11
Fig. 11 Pump efficiency with different wall roughness levels under varying flow conditions (η 0 is the efficiency of the smooth wall)

Table 1
Main design parameters of the dredge pump

Table 2
Conversion between Ra and K s