# Effect of Velocity Slip on Head Prediction for Centrifugal Pumps as Turbines

Authors: Xiaohui Wang; Junhu Yang; Zhengting Xia; Yan Hao; Xiaorui Cheng
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The application of pumps as turbines (PAT) has been developed in several applications for energy recovery schemes. Therefore, establishing a performance correlation between pump mode and turbine mode is essential for selecting the proper machine. However, slip phenomenon is the challenges of head prediction for PAT. In this paper, the slip phenomenon of pump and PAT was revealed, and the slip factor was studied using CFD. The effect of slip on head prediction for PAT was analyzed, and a theoretical prediction model was presented considering slip factors. In order to validate the head prediction model, six centrifugal pumps with specific speed (ns) from 9 to 54.8 were tested as turbines. Results showed that the predicted head by the proposed method was in good agreement with the experimental data, and it is more accurate than Stepanoff, Alatorre-Frenk, Sharma, and Derakhshan models. This method can be applied in head prediction for low specific speed PAT (ns <60).


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## Body

## 1. Introduction

Centrifugal pumps operating as turbines (PAT) are an attractive and important alternative for energy recovery solution. The use of PAT has been adapted in remote area power supply installations, as well as industrial application in energy recovery systems. Compared with traditional micro-hydro turbines, the PAT is a relatively simple machine, available for a wide range of heads and flows, easy to maintain, and readily available all over the world [1].In recent studies, researches have attempted to build a model that would make accurate predictions of the turbine operation of pumps at best efficiency points (BEP). There have been several prediction techniques published so far, as reported by Stepanoff [2], Alatorre-Frenk [3], Sharma [4], Williams [5], Singh [6], Derakhshan, and Nourbakhsh [7, 8], Wang [9], and Ramos [10]. Most of these studies have suggested the relations for prediction of PAT behavior either based on efficiency or based on specific speed in pump mode, and a few have also derived the relations based on experimentation. However, the results obtained from these studies had an approximately ±20% deviation from experimental data [5], with deviations exceeding over 40% for some specific speeds [11]. One major reason for this was the slip phenomenon in turbine mode has not been considered, which has prevented an accurate performance prediction of PAT [12].Slip phenomenon is known to strongly affect the performance of PAT, especially at off-design condition; it is usually considered as an important factor in the design process of turbo-machines (pump, compressor, or turbine) [13, 14]. In order to predict the performance characteristics, velocity slip within flow passages should be considered both in pump and in turbine modes, and the slip factor was calculated by means of numerical simulation in their papers [15, 16]. When a pump is running as a turbine, because of the velocity slip within impeller flow passages, the theoretical head of turbine is lower than Euler head, where the Euler head was obtained based on the infinite-blade hypothesis. Velocity slip should be taken into account accurately for head prediction of PAT. As a consequence, it is important to investigate slip factor and reveal the effect of slip on head prediction for PAT.In order to overcome the challenges of head prediction for PAT, the velocity slip of PAT was researched and slip factor was calculated. Then a new head prediction model is proposed in consideration of slip factor in pump and turbine mode. Six centrifugal pumps with specific speed (ns) from 9 to 54.8 were tested as turbines, and the head of the proposed model was compared with Stepanoff [2], Alatorre-Frenk [3], Sharma [4], and Derakhshan [8] methods. Deviations from experimental data are presented.


## 2. Theoretical Analysis

When a centrifugal pump operates as a turbine, flow and rotating directions are reverse. The velocity distribution within the impeller is shown in Figure1 and heavy triangles are based on theoretical blade-to-blade analysis (Euler infinite-blade assumption), while dashed triangles are realistic velocity, where subscripts 1 and 2 refer to the high pressure side and low pressure side of the blade, respectively.Figure 1
Velocity triangle of impeller.
(a)
Pump mode (b)
Turbine modeDue to the fact that the blades of PAT are always finite, the realistic flow in impeller passage is produced by axial-eddy and through flow [17, 18], which causes a difference between the flow angleβ’ and the blade angle β. This angular difference corresponds to an absolute-tangential velocity difference Δvu (Figure 1). This is known as slip velocity, which is different in pump mode and turbine mode. For pump, the relative velocity at the outlet (high pressure side) of impeller does not match the blade surface, inducing slip velocity Δvu1, when a pump runs as turbine. Similarly, the slip velocity occurs at the inlet of impeller, as shown in Figure 2. Additionally, slip velocity also exists at pump blade inlet and turbine blade outlet, but it was neglected in this paper since the Δvu2 is very small. The slip velocity induces the change of vu1, which causes the theoretical head of impeller to be smaller than Euler head. Therefore, slip phenomenon should be considered for head prediction of PAT.Figure 2
Slip phenomenon for pump and turbine modes.If the rotational speeds are the same, there are equal and opposite heavy line velocities for pump mode and turbine modes based on the infinite-blade theory. Consequently, Euler head in pump and turbine mode are the same(1)HEuler=u1vu1-u2vu2gwhere u is peripheral speed, vu is peripheral component of velocity,g is gravitational acceleration, and subscripts 1 and 2 devote the high pressure side and low pressure side of the blade, respectively.In (1), vu2 is negligibly small in general; as a result, the Euler head can be as(2)HEuler=u1vu1gwhere u1 and vu1 can be obtained depend on the velocity triangle as shown in Figure 1(b) with heavy line,(3)u1=πnD160(4)vu1=u1-QπD1b1cot⁡βb1where n is impeller rotational speed, D is impeller diameter, b is flow passage width, βb is blade angle, and Q is flow rate.Because of the finite-blade of impeller, the theoretical headHth of impeller is difficult to obtained by theoretical method, which can be predicted by CFD as follows:For pump mode(5)Hth.p=HpηhpFor turbine mode(6)Hth.t=Ht·ηhtwhere H is net head and ηh is hydraulic efficiency of impeller.Therefore, the head reductionΔH caused by slip velocity for pump mode and turbine mode is (7)ΔHp=HEuler-Hth.p(8)ΔHt=HEuler-Hth.tThen, the slip factor of pump and PAT can be described as (9)sp=ΔHpHEuler(10)st=ΔHtHEulerAs mentioned above, the Euler head HEuler can be calculated using (1); theoretical head Hth can be obtained by Computational Fluid Dynamics (CFD) methodology; as a result, the slip factor was acquired by (9) and (10).In this paper, the slip factor of pump and turbine modes was studied. The effect of velocity slip on head prediction of PAT was analyzed, and a prediction model was presented. Finally, an experiment was performed to verify the prediction model.


## 3. Numerical Investigation

### 3.1. Numerical Simulation

In the present work, CFD was used for numerical research of PAT. Considering the Direct Numerical Simulation (DNS) is out of range due to the high computer power needed, and Large Eddy Simulation (LES) is currently only feasible for simplified pump elements [19], the computations were performed by means of the Reynolds Average Navier-Stokes (RANS) equations with an appropriate turbulence model. In this study, the ANSYS-CFX was selected for the solution of the steady 3D Navier-Stokes equation. There are many turbulence models in use today, and four of the most popular turbulence models are the standardk-ε, RNGk-ε, standardk-ω, and SSTk-ωmodel. In the attempt to adopt the most appropriate turbulence model for solving RANS equations, these four models were applied, respectively, for PAT performance simulation. The head of PAT calculated by these four models and experiment data was compared, and the most accurate one was chosen.A centrifugal pump with specific speed 9.0 is operating as a turbine for numerical simulation (Figure3). The head, discharge, and rotational speed of selected pump were 32 m, 25m3/h, and 1,450 rpm, respectively. Table 1 gives the main geometry parameters of the PAT. The 3D model of this PAT was generated including the volute, impeller, and draft tube.Table 1
The main geometric parameters of PAT.
category parameter Impeller D 1(mm) 312 D 2(mm) 80 d h(mm) 0 b 1(mm) 10 β 1(°) 32 Z 6 θ(°) 150  Volute D i n (mm) 50 b 0 (mm) 24 D 0 (mm) 320  Draft tube D o u t (mm) 80Figure 3
The geometrical model of PAT.Considering the numerical simulation converges more rapidly and accurately by utilizing structure grid, an ICEM-CFD was used to generate a pure hexahedral grid for each component part of simulated domain (Figure4). The different y+ should be satisfied for different turbulence models. For thek-ε turbulence model, a near wall result must be obtained using scalable wall function, for a y+ near 200, while y+ ≤100 was appropriate for thek-ω model with automatic wall function [20]. The range of y+ for near wall is provided in Table 2.Table 2
Range of Y+ for near-wall mesh.
Standardk-ε RNGk-ε Standardk-ω SSTk-ω Volute 7.6~184.2 7.6~184.2 5.2~87.5 5.2~87.5 Impeller 3.7~194.3 3.7~194.3 2.3~98.7 2.3~98.7 Draft tube 26.3~31.4 26.3~31.4 26.3~31.4 26.3~31.4Figure 4
Structured grid of PAT.A grid sensitivity analysis was performed to verify the grid-independence of results. It was found that when the grid element was beyond 4×106, the heads of PAT were almost constant regardless of using standardk-ε, RNGk-ε, standardk-ω, and SSTk-ω model (Figure 5). The final elements number of fluid volume was 4,154,084.Figure 5
Grid independent test.


### 3.2. Validation of Numerical Simulation

The PAT was simulated and the results were compared with experimental data. The experimental test rig was showed in Figure6. A feed pump was installed to provide the pressure flow for PAT, and another pump was connected to PAT shaft for consuming the energy generated by PAT. A flow regulating valve was installed at outlet pipe of consume pump for keeping the rotational speed. Two pressure transmitters ABG-M2 with range of 0-2MPa were set at the inlet and outlet of PAT, respectively, for measuring pressure, an electromagnetic flow meter MF200-101 was set at the inlet pipe of PAT for measuring the discharge, and a NJ1 torque meter was installed at PAT shaft for measuring the rotational speed and torque of PAT. The discharge, head, shaft power, and efficiency of PAT were calculated after all parameters were measured. The uncertainty in different performance parameters was estimated in accordance with the procedure as described by Moffat [21]. The relative uncertainty in discharge, head, shaft power, and efficiency of PAT were ±0.82%, ±1.2%, ±1.72%, and ±2.5%, respectively.Figure 6 (a)
Schematic diagram of experimental test rig (b)
Test rig of PATSimulation using four turbulence models mentioned above was done for tested PAT, and the validation results were shown in Figure7. CFD could not simulate disc friction, mechanical losses, and leakage since the flow volume did not include the space between impeller hub/shroud and casing or the sealing gap. Despite the loss generated in experiment of PAT, the PAT need more head to overcome the loss reasonably. As a result, the head of PAT predicted by CFD was slightly lower than the experimental result. This was confirmed by other papers [7, 22].Figure 7
CFD and experimental results of PAT.Here,Φ and Ψ are flow rate and head coefficient, respectively, which can be defined as(11)ϕ=QnD13(12)ψ=gHn2D12The results showed that the mean deviations of standardk-ε, RNGk-ε, standardk-ω, and SSTk-ω model from the experimental result were 6.62%, 5.53%, 7.82%, and 6.46%, while the mean-square deviations were 0.63%, 0.33%, 0.56%, and 1.65%, respectively. It can be seen that the minimum deviation was the RNGk-ε model. Therefore, the RNGk-ε model confirms a good accuracy in the performance prediction of PAT; it was adopted for head prediction in this paper.


## 3.1. Numerical Simulation

In the present work, CFD was used for numerical research of PAT. Considering the Direct Numerical Simulation (DNS) is out of range due to the high computer power needed, and Large Eddy Simulation (LES) is currently only feasible for simplified pump elements [19], the computations were performed by means of the Reynolds Average Navier-Stokes (RANS) equations with an appropriate turbulence model. In this study, the ANSYS-CFX was selected for the solution of the steady 3D Navier-Stokes equation. There are many turbulence models in use today, and four of the most popular turbulence models are the standardk-ε, RNGk-ε, standardk-ω, and SSTk-ωmodel. In the attempt to adopt the most appropriate turbulence model for solving RANS equations, these four models were applied, respectively, for PAT performance simulation. The head of PAT calculated by these four models and experiment data was compared, and the most accurate one was chosen.A centrifugal pump with specific speed 9.0 is operating as a turbine for numerical simulation (Figure3). The head, discharge, and rotational speed of selected pump were 32 m, 25m3/h, and 1,450 rpm, respectively. Table 1 gives the main geometry parameters of the PAT. The 3D model of this PAT was generated including the volute, impeller, and draft tube.Table 1
The main geometric parameters of PAT.
category parameter Impeller D 1(mm) 312 D 2(mm) 80 d h(mm) 0 b 1(mm) 10 β 1(°) 32 Z 6 θ(°) 150  Volute D i n (mm) 50 b 0 (mm) 24 D 0 (mm) 320  Draft tube D o u t (mm) 80Figure 3
The geometrical model of PAT.Considering the numerical simulation converges more rapidly and accurately by utilizing structure grid, an ICEM-CFD was used to generate a pure hexahedral grid for each component part of simulated domain (Figure4). The different y+ should be satisfied for different turbulence models. For thek-ε turbulence model, a near wall result must be obtained using scalable wall function, for a y+ near 200, while y+ ≤100 was appropriate for thek-ω model with automatic wall function [20]. The range of y+ for near wall is provided in Table 2.Table 2
Range of Y+ for near-wall mesh.
Standardk-ε RNGk-ε Standardk-ω SSTk-ω Volute 7.6~184.2 7.6~184.2 5.2~87.5 5.2~87.5 Impeller 3.7~194.3 3.7~194.3 2.3~98.7 2.3~98.7 Draft tube 26.3~31.4 26.3~31.4 26.3~31.4 26.3~31.4Figure 4
Structured grid of PAT.A grid sensitivity analysis was performed to verify the grid-independence of results. It was found that when the grid element was beyond 4×106, the heads of PAT were almost constant regardless of using standardk-ε, RNGk-ε, standardk-ω, and SSTk-ω model (Figure 5). The final elements number of fluid volume was 4,154,084.Figure 5
Grid independent test.


## 3.2. Validation of Numerical Simulation

The PAT was simulated and the results were compared with experimental data. The experimental test rig was showed in Figure6. A feed pump was installed to provide the pressure flow for PAT, and another pump was connected to PAT shaft for consuming the energy generated by PAT. A flow regulating valve was installed at outlet pipe of consume pump for keeping the rotational speed. Two pressure transmitters ABG-M2 with range of 0-2MPa were set at the inlet and outlet of PAT, respectively, for measuring pressure, an electromagnetic flow meter MF200-101 was set at the inlet pipe of PAT for measuring the discharge, and a NJ1 torque meter was installed at PAT shaft for measuring the rotational speed and torque of PAT. The discharge, head, shaft power, and efficiency of PAT were calculated after all parameters were measured. The uncertainty in different performance parameters was estimated in accordance with the procedure as described by Moffat [21]. The relative uncertainty in discharge, head, shaft power, and efficiency of PAT were ±0.82%, ±1.2%, ±1.72%, and ±2.5%, respectively.Figure 6 (a)
Schematic diagram of experimental test rig (b)
Test rig of PATSimulation using four turbulence models mentioned above was done for tested PAT, and the validation results were shown in Figure7. CFD could not simulate disc friction, mechanical losses, and leakage since the flow volume did not include the space between impeller hub/shroud and casing or the sealing gap. Despite the loss generated in experiment of PAT, the PAT need more head to overcome the loss reasonably. As a result, the head of PAT predicted by CFD was slightly lower than the experimental result. This was confirmed by other papers [7, 22].Figure 7
CFD and experimental results of PAT.Here,Φ and Ψ are flow rate and head coefficient, respectively, which can be defined as(11)ϕ=QnD13(12)ψ=gHn2D12The results showed that the mean deviations of standardk-ε, RNGk-ε, standardk-ω, and SSTk-ω model from the experimental result were 6.62%, 5.53%, 7.82%, and 6.46%, while the mean-square deviations were 0.63%, 0.33%, 0.56%, and 1.65%, respectively. It can be seen that the minimum deviation was the RNGk-ε model. Therefore, the RNGk-ε model confirms a good accuracy in the performance prediction of PAT; it was adopted for head prediction in this paper.


## 4. Results and Discussion

### 4.1. Slip Phenomenon in Pump and PAT

Figure8 gives the streamline of pump and PAT at best efficiency point; it can be seen that axial vortex occurred within flow passages. As predicted in Figure 2, the relative velocity did not match to the blade surface owing to the flow slip and the axial vortex was derived near the pressure side of blade. The vortex direction was opposite to impeller rotation, and the vortex structure and position were different for pump and PAT; it occurred next to the center of blade and was surrounded with main flow for pump and occurred near the outlet of blade for PAT.Figure 8
Streamline of impeller flow.
(a)
Pump mode (b)
Turbine modeThe uniform flow distribution would be induced by axial vortex, which influences the energy transformation of blades reasonably. Figure9 gives the relative velocity distribution on the blade of pump and PAT. It was observed that the relative velocity increased uniformly along the blade of suction side for both pump and PAT, and humped (increased and then decreased) for pressure side, which was caused by slip phenomenon within flow passages. The bad velocity distribution leads to the pressure fluctuate on the blade, as shown in Figure 10. The pressure distribution of pump was more uniform than PAT since the vortex was surrounded and difficult to spread; however, pressure fluctuations can be observed near the outlet of blade for PAT due to the unstable vortex, which was derived next to blade outlet and interacted with flow in volute. Additionally, the nonuniform velocity distribution in flow passage induces the friction of adjacent flow particle, which leads to the additional hydraulic loss.Figure 9
Relative velocity distribution of PAT impeller.
(a)
Pump mode (b)
Turbine modeFigure 10
Pressure distribution of PAT impeller.
(a)
Pump mode (b)
Turbine modeSlip phenomenon induces the axial vortex next to the blade surface, the vortex regions are always the low pressure region, and additionally, pressure fluctuation can be observed. As a consequence, the energy transformed between fluid and blade is reduced in these regions, the theoretical head smaller than Euler head positively. This phenomenon exists both in pump and in PAT, but its behavior is different from each other. Therefore, the slip phenomenon should be considered in performance prediction of PAT.


### 4.2. Slip Factor

The slip factor of pump and PAT is different due to opposite flows. It is of interest to investigate the slip phenomenon and to obtain the effective value of slip factor of pump and turbine mode. As mentioned in (7)–(10), the slip factor is related to the Euler head and the theoretical head. The Euler head was obtained by theoretical calculation using (1) for each PAT, and the theoretical head was achieved using CFD simulation in this paper. As a result, the slip factor was obtained.Figure11 shows the slip factor and hydraulic efficiency for pump mode and turbine mode of PAT in all examined conditions. It was observed that the slip factors of pump and turbine modes were different in both design condition and off-design condition. The slip factor increased with flow rate in pump operation and decreased in turbine operation. The value approached 0.28 at BEP for pump mode and 0.24 for turbine mode. It rose dramatically at low flow rate of turbine mode, and turbine efficiency decreased similarly. The larger the S value, the more serious the slip. From the theoretical point of view, slip phenomenon induces additional hydraulic loss in impeller passage of PAT. The slip phenomenon strengthened in part-load region; thus the hydraulic efficiency was relatively low. On the contrary, the slip phenomenon weakened in the over load region, so the hydraulic efficiency of PAT was relatively high. This is also a reason why PAT’s efficiency is higher in large flow rate. Therefore, when a centrifugal pump runs as a turbine, it is not recommended for the machine to work at a low flow rate.Figure 11
Slip factor and efficiency curves of PAT.For investigating the slip factor of pump and turbine modes at BEP, six centrifugal pumps with specific speeds from 9 to 54.8 were selected. The geometric parameters of these pumps are shown in Table3. Figure 12 shows that the slip factor varied with different specific speed of PAT, and it was higher in pump mode than turbine mode at BEP; the hydraulic efficiency of the pump mode was lower than turbine mode.Table 3
Geometric parameters of PATs.
ns Impeller parameters Volute parameters D 1(mm) D 2(mm) b 1(mm) β b 1(°) θ(°) D 0(mm) b 0(mm) D i n(mm) 9.0 312 80 10 32 150 320 24 80 15.1 272 98 10 32 138 280 26 80 23.6 210 90 16 26 134 220 36 65 35.1 216 125 26.5 17/22 107 230 42 100 49.3 143 86 24 29 86 160 40 80 54.8 246 157 39 26 100 256 60 200Figure 12
Slip factor and hydraulic efficiency curves.


### 4.3. Performance Prediction of PAT

Using (5)–(10), the net head of pump and PAT can be described as(13)Hp=ηhp·1-sp·HEuler(14)Ht=1-st·HEulerηhtTherefore, at the same rotational speed, the head correlation between pump and PAT is(15)HtHp=1-st1-sp1ηhtηhpSix industrial centrifugal pumps with specific speeds from 9 to 54.8 were tested and the results are shown in Table 4. The predictions obtained by Stepanoff, Alatorre-Frenk, Sharma, Derakhshan, and the proposed method are listed in the table for comparison.Table 4
Head correlations between pump and PAT.
n s H t / H p Stepanoff Alatorr-Frenk Sharma Derakhshan New method Experimental data 9.0 1.42 2.32 2.01 1.93 1.58 1.55 15.1 1.98 1.93 1.56 2.04 1.81 1.63 23.6 1.72 1.47 1.29 1.83 1.49 1.54 35.1 1.62 1.41 1.26 1.64 1.47 1.60 49.3 1.70 1.31 1.22 1.45 1.61 1.53 54.8 1.40 1.22 1.17 1.38 1.52 1.48Figure13 shows the deviations of these methods from the experimental data. The slip phenomenon was considered in the proposed new method of head prediction; it can be seen that the mean deviations of predicted head correlation by Stepanoff, Alatorre-Frenk, Sharma, Derakhshan and the suggested new method were 8.60%, 16.72%, 15.23%, 13.83%, and 5.70% from experimental data, respectively. The results showed that the proposed method had the minimum mean deviation, which means this method closely resembles the experimental result. Thus it can be considered that the proposed method is more accurate for PAT with 9<ns<54.8 than the others. This method can be applied in performance prediction for pump operating as turbine.Figure 13
Head deviations based on experimental data.


## 4.1. Slip Phenomenon in Pump and PAT

Figure8 gives the streamline of pump and PAT at best efficiency point; it can be seen that axial vortex occurred within flow passages. As predicted in Figure 2, the relative velocity did not match to the blade surface owing to the flow slip and the axial vortex was derived near the pressure side of blade. The vortex direction was opposite to impeller rotation, and the vortex structure and position were different for pump and PAT; it occurred next to the center of blade and was surrounded with main flow for pump and occurred near the outlet of blade for PAT.Figure 8
Streamline of impeller flow.
(a)
Pump mode (b)
Turbine modeThe uniform flow distribution would be induced by axial vortex, which influences the energy transformation of blades reasonably. Figure9 gives the relative velocity distribution on the blade of pump and PAT. It was observed that the relative velocity increased uniformly along the blade of suction side for both pump and PAT, and humped (increased and then decreased) for pressure side, which was caused by slip phenomenon within flow passages. The bad velocity distribution leads to the pressure fluctuate on the blade, as shown in Figure 10. The pressure distribution of pump was more uniform than PAT since the vortex was surrounded and difficult to spread; however, pressure fluctuations can be observed near the outlet of blade for PAT due to the unstable vortex, which was derived next to blade outlet and interacted with flow in volute. Additionally, the nonuniform velocity distribution in flow passage induces the friction of adjacent flow particle, which leads to the additional hydraulic loss.Figure 9
Relative velocity distribution of PAT impeller.
(a)
Pump mode (b)
Turbine modeFigure 10
Pressure distribution of PAT impeller.
(a)
Pump mode (b)
Turbine modeSlip phenomenon induces the axial vortex next to the blade surface, the vortex regions are always the low pressure region, and additionally, pressure fluctuation can be observed. As a consequence, the energy transformed between fluid and blade is reduced in these regions, the theoretical head smaller than Euler head positively. This phenomenon exists both in pump and in PAT, but its behavior is different from each other. Therefore, the slip phenomenon should be considered in performance prediction of PAT.


## 4.2. Slip Factor

The slip factor of pump and PAT is different due to opposite flows. It is of interest to investigate the slip phenomenon and to obtain the effective value of slip factor of pump and turbine mode. As mentioned in (7)–(10), the slip factor is related to the Euler head and the theoretical head. The Euler head was obtained by theoretical calculation using (1) for each PAT, and the theoretical head was achieved using CFD simulation in this paper. As a result, the slip factor was obtained.Figure11 shows the slip factor and hydraulic efficiency for pump mode and turbine mode of PAT in all examined conditions. It was observed that the slip factors of pump and turbine modes were different in both design condition and off-design condition. The slip factor increased with flow rate in pump operation and decreased in turbine operation. The value approached 0.28 at BEP for pump mode and 0.24 for turbine mode. It rose dramatically at low flow rate of turbine mode, and turbine efficiency decreased similarly. The larger the S value, the more serious the slip. From the theoretical point of view, slip phenomenon induces additional hydraulic loss in impeller passage of PAT. The slip phenomenon strengthened in part-load region; thus the hydraulic efficiency was relatively low. On the contrary, the slip phenomenon weakened in the over load region, so the hydraulic efficiency of PAT was relatively high. This is also a reason why PAT’s efficiency is higher in large flow rate. Therefore, when a centrifugal pump runs as a turbine, it is not recommended for the machine to work at a low flow rate.Figure 11
Slip factor and efficiency curves of PAT.For investigating the slip factor of pump and turbine modes at BEP, six centrifugal pumps with specific speeds from 9 to 54.8 were selected. The geometric parameters of these pumps are shown in Table3. Figure 12 shows that the slip factor varied with different specific speed of PAT, and it was higher in pump mode than turbine mode at BEP; the hydraulic efficiency of the pump mode was lower than turbine mode.Table 3
Geometric parameters of PATs.
ns Impeller parameters Volute parameters D 1(mm) D 2(mm) b 1(mm) β b 1(°) θ(°) D 0(mm) b 0(mm) D i n(mm) 9.0 312 80 10 32 150 320 24 80 15.1 272 98 10 32 138 280 26 80 23.6 210 90 16 26 134 220 36 65 35.1 216 125 26.5 17/22 107 230 42 100 49.3 143 86 24 29 86 160 40 80 54.8 246 157 39 26 100 256 60 200Figure 12
Slip factor and hydraulic efficiency curves.


## 4.3. Performance Prediction of PAT

Using (5)–(10), the net head of pump and PAT can be described as(13)Hp=ηhp·1-sp·HEuler(14)Ht=1-st·HEulerηhtTherefore, at the same rotational speed, the head correlation between pump and PAT is(15)HtHp=1-st1-sp1ηhtηhpSix industrial centrifugal pumps with specific speeds from 9 to 54.8 were tested and the results are shown in Table 4. The predictions obtained by Stepanoff, Alatorre-Frenk, Sharma, Derakhshan, and the proposed method are listed in the table for comparison.Table 4
Head correlations between pump and PAT.
n s H t / H p Stepanoff Alatorr-Frenk Sharma Derakhshan New method Experimental data 9.0 1.42 2.32 2.01 1.93 1.58 1.55 15.1 1.98 1.93 1.56 2.04 1.81 1.63 23.6 1.72 1.47 1.29 1.83 1.49 1.54 35.1 1.62 1.41 1.26 1.64 1.47 1.60 49.3 1.70 1.31 1.22 1.45 1.61 1.53 54.8 1.40 1.22 1.17 1.38 1.52 1.48Figure13 shows the deviations of these methods from the experimental data. The slip phenomenon was considered in the proposed new method of head prediction; it can be seen that the mean deviations of predicted head correlation by Stepanoff, Alatorre-Frenk, Sharma, Derakhshan and the suggested new method were 8.60%, 16.72%, 15.23%, 13.83%, and 5.70% from experimental data, respectively. The results showed that the proposed method had the minimum mean deviation, which means this method closely resembles the experimental result. Thus it can be considered that the proposed method is more accurate for PAT with 9<ns<54.8 than the others. This method can be applied in performance prediction for pump operating as turbine.Figure 13
Head deviations based on experimental data.


## 5. Conclusion

In the present study, the velocity slip of pump and PAT was revealed, and the slip factor was studied using CFD. The effect of velocity slip on head prediction for PAT was analyzed, and a theoretical prediction model was presented considering slip factors. Finally, an experiment was performed to verify the prediction model. The results showed that the slip factor of pump was higher than PAT at BEP. However, the hydraulic efficiency was smaller at BEP.Six centrifugal pumps with specific speed from 9 to 54.8 were tested as turbines. Results showed that the predicted head by the proposed method was in good agreement with the experimental data. Additionally, the predictions of the proposed method were more accurate than the predictions of Stepanoff, Alatorre-Frenk, Sharma, and Derakhshan models for PAT with9<ns<54.8.



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