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Latin American applied research

On-line version ISSN 1851-8796

Lat. Am. appl. res. vol.44 no.1 Bahía Blanca Jan. 2014


Numerical calculation and experimental study of axial force in a deep-well centrifugal pump

L. Zhou, W.D. Shi, L. Bai, W.G. Lu and W. Li

National Research Center of Pumps & Pumping System Engineering & Technology, Jiangsu University, China.

Abstract — In the operation process of centrifugal pumps, especially in multi-stage pumps, axial force is one of the main factors which affect the pump safety and reliability. This paper presents the axial force study in a deep-well centrifugal pump (DCP) with theoretical calculation, numerical simulation and experimental measure. Three different calculation formulas were respectively introduced and used in the model pump. The calculated results were compared and analyzed with the numerical simulation values and experimental results, and the detailed numerical simulation methods and experimental configuration were explained. Finally, the more accurate formula for calculating the axial force in oblique flow centrifugal impeller was selected out. At the rated flow point, the deviation of the axial force obtained by numerical simulation and the experimental value is approximately 3.8%, and the calculated result of selected formula only less than the experimental results of 2.6%. The results provide a theoretical basis for the axial force calculation in the centrifugal impeller design process.

Keywords — Numerical Simulation; Deep-Well Centrifugal Pump; Centrifugal Impeller; Axial Force.


One of the most challenging and difficult aspects in multistage pumps design is represented by the calculation and balance of the axial force acting on the rotating shaft, especially in high-head deep-well centrifugal pumps (DCPs)(Childs, 1991; Guelich, 2007). Although the balance device such as bearings is normally installed on the multistage pumps, but due to the inaccurate calculation of axial force, the balance equipment still has the possibility of damage in the pump operating range, and in severe cases even result in the breakdown of the pump system. In other words, the calculation of the axial force which acts on the rotor during operation will directly affect the pump safety and reliability. Consequently, it is important to obtain accurate estimates of axial force for proper selection of thrust bearings, operating condition, and pump design (González et al., 2006; Schaefer and Olson, 1999).

Several research works have been published during the past decades, concerning axial force studies on radial flow centrifugal pumps. Kazakov and Pelinskii (1970) carried out the experimental investigation of the axial force in a submersible well pump, which demonstrated the axial force in the pump increases with the increases of seal clearance. Lino et al. (1980) described an experimental and theoretical study with a single-stage pump, and a computer program for calculating the axial force in multistage centrifugal pumps was developed. Guelich et al. (1987) discussed the two axial force prediction methods and compared these results on a multistage boiler feed pump. Gantar et al. (2002) applied the Laser Doppler Anemometry (LDA) and numerical flow analysis method to study the axial force in multistage pump, and the results show the possibility that the impeller side chambers can be affected and the hydraulic axial force can be reduced.

Deep-well centrifugal pumps (DCPs), the main equipment for pumping underground water, have been widely used in rural area, field irrigation, as well as geothermal utilization. The increasing performance levels and operating conditions requirements make the task of designing a DCP very challenging. Besides, the construction of high pressure DCPs requires a solution of the axial force produced by the pump operation.

Our previous work proposed a new type of DCP (Shi et al., 2009), as shown in Fig.1. In order to increase the single-stage head, the diameter of the impeller front shroud is almost increased to its maximum value. The impeller rear shroud and the impeller blade outlet side are beveled in the axial direction. Compared with the similar products in the domestic and foreign markets, the single-stage head of this new DCP has been increased by 20%~50%, and the pump efficiency has been increased by 3%~9%. Besides, the single-stage axial length and the weight of this new type DCP have been reduced by about 1/3, showing good energy and material savings features. This type oblique flow centrifugal impeller is widely used in DCP in recent years, but the research for calculation method of its axial force is still rarely.

Fig. 1. Assembly drawing of new type DCP

The present study aims to investigate the axial force in new type DCP, and explore the accurate method for axial force calculation. Consequently, with the integration of theoretical analysis, numerical simulation and experimental results, three different calculation formulas of axial force were compared and analyzed, and a detailed numerical simulation and an axial force experiment were carried out to verify these results.


This paper chose the 150QJ20-65/5 type DCP as the research model pump, and its main parameters at the design condition are as follows: design flow rate Qd=20m3/h, single-stage head Hs=11m, speed n=2850 r/min, specific speed ns= 35.

The impeller's cross-section and solid model created using Pro/E are shown in Fig.2. The detailed structural parameters of impeller are present in Table.1.

(a) Cross-section

(b) Solid model
Fig. 2. Impeller

Table 1 Geometric parameters of impeller


Figure 3 is the structure diagram of oblique flow centrifugal impeller.

Fig. 3. Structure diagram of oblique flow centrifugal impeller

The collected calculation formulas of axial force in oblique flow impeller are listed in the following. In these formulas, the gravity of impeller and shaft are not counted. Besides, these formulas assume that the impeller center line coincides with the center line of guide vane, and this coincidence does not change in the pump operating range

(1) Chinese pump researcher Guan (1995) recommends the calculation formula of axial force is as follows:


where F is the total axial force; Fd is the dynamic reaction force; Hp is the potential head in impeller outlet; ω is the impeller angular velocity; Qt is the pump theoretical flow rate; rh is the radius of hub; rm is the radius of seal ring; r2a is the radius of impeller front shroud; r2i is the radius of impeller rear shroud; α is the angle between impeller exit axial velocity and axis; vm0 is the axial velocity slightly before impeller blade inlet; and vm2 is the axial velocity slightly later impeller blade outlet;

(2) Pfleiderrer (1964) proposed the following formula to calculate axial force of oblique flow impeller:

F = F1i - F1a + Fp - Fd (3)

where , the value of axial force on impeller rear shroud; , the value of axial force on impeller front shroud; , the value of axial direction component of gap pressure; Hpi is the potential head at impeller outside diameter of black cover plate; and Hpa is the potential head at impeller outside diameter of front cover plate.

(3) The above two formulas have the same assumptions that angular velocity of high-pressure liquid outside the impeller seal ring is the half of the shroud angular velocity, the pressure distribution according to parabolic law along the radial. With a large number of comparisons between the calculated value and measured value of axial force in different impellers, the author Lu et al. (2011) has amended the value of circumferential velocity around the impeller. In his opinions, the liquid circumferential velocity in the lateral of the impeller shroud is 0.45ω, and the liquid circumferential velocity in the oblique side of impeller outlet is 0.65ω. The amended formula for calculating axial force is as follows:


where , the value of axial force on underside of impeller front shroud; , the value of axial force on impeller rear shroud; , the value of axial force on upside of impeller front shroud; define x and y as the correction coefficient of circumferential velocity, and x=0.65, y=0.45.

The above three formulas were used to calculate the axial force of 150QJ20-65/5 type DCP. At the rated flow point, the formula calculation results were indicated in Table 2. These formulas were based on different assumptions, so the above formula calculation results vary greatly. There is no report about that using which formula calculation result could be consistent with measured results. It is necessary to combine numerical simulation and experimental methods to verify the accuracy of these formulas.

Table 2 Calculation results with three formulas


A. Calculation model

The whole flow domain of two stages DCP consists of 5 components: inlet section, seal ring, impeller, guide vane, and outlet section, as shown in Fig. 4. After modeled in Pro/E, the assembly model was imported to Gambit for a further processing.

Fig. 4. Sketch of flow domain

B. Mesh and grid sensitivity analysis

The whole mesh generation process was carried out in the Gambit software, in addition to the inlet section, seal ring and outlet section were meshed with structured girds, and the other components were meshed with unstructured grids. In this paper, five different grid numbers were used to carry out the numerical simulation, and the results were indicated in Table 3.

Table 3 Grid sensitively analysis

Table 3 shows the results of grid sensitivity analysis, it can be noted that when the grids size is less than 1.4 or grids number is larger than 1.58 million, the single-stage head and the efficiency only experience slight change, indicating that the numerical simulation results are becoming stable. Considering the computer's calculation capability, the grids size is selected as 1.4 to carry out the following study.

C. Boundary conditions

The whole hydraulic passage of the two stages deep-well centrifugal pump was taken as the computational flow domain. The impeller passages are attached to the rotating frame and solved in a rotating framework via the Multiple Reference Frame (MRF), and the rotational speed was set as 2850 rpm. The interfaces were formed between the different regions.

A uniform axial velocity based on the mass-flow rate is specified at the inlet, and the outlet boundary was assumed to be outflow. So at the outlet, which is roughly two impeller-diameters downstream of the vane trailing edge, the gradients of the velocity components are assumed to be zero. At the exit pipes, there is an unavoidable effect on the final flow solution as a result of the boundary conditions. A reasonable length added to the real machine geometry to avoid this effect as much as possible. This paper select the grids size of 1.4, and the value of y+ is 137, which indicates the near-wall nodes are not within the laminar sub-layer but within the log-low layer (Storti et al., 2011). So the standard wall function was approached to the turbulent flow of near-wall, and all physical surfaces of the pump were set to be no-slip wall.

D. Turbulence model

There is no universally valid turbulence model which will yield optimum results for all applications. Instead it is necessary to select the turbulence model most suitable for the components to be calculated and to carefully validate it by comparing the numerical results with test data. Six turbulence models were adopted to carry out the numerical flow calculations, namely, standard k-ε model, RNG k-ε model, realizable k-ε model, standard k-ω model, SST k-ω model, and Spalart-Allmaras model.

Table 4 compares the test and numerical results with different turbulence models. All of the numerical results of five turbulence models are higher than experiment value, among which the results of standard k-ε model are closest to the experiment data. Thus the standard k-ε model is chosen for the following numerical calculation.

Table 4 Numerical results with different turbulence models

E. Numerical algorithm and other parameters

The flow through the modeled pump was simulated with the commercial code Fluent, which uses the finite volume method to solve the Reynolds averaged Navier-Stokes equations for 3D incompressible steady flow. The flow model was complemented with a standard k-ε model and logarithmic-law functions for the near wall flow, consistent with the non-slip wall condition. Second order upwind discretizations were used for the convective and the diffusive terms. The time dependent term scheme was second order implicit. The pressure-velocity coupling was calculated by means of the SIMPLEC algorithm, and the convergence precision is set to 10-5. During the numerical study, the guidelines proposed in reference were used and the numerical un certainty was related to the change in certain reference values when different mesh refinements were consid ered. In other words, the values obtained for accuracy can be considered reasonable as to validate the numerical results.

F. The results of axial force

The summation of forces in axial direction that stressed on solid surface of impeller just is the axial force. As shown in Fig.5, in order to calculate the axial force precisely from the numerical results, the total axial force were divided to six components:

F = F1 + F2 + F3 + F4 + F5 + F6 (5)

where F1 is the axial force on outside surfaces of impeller front shroud, F2 is the axial force on outside surfaces of impeller rear shroud, F3 is the axial force on circular region of seal ring, F4 is the axial force of impeller inner surfaces, F5 is the axial force of blade surfaces, and F6 is the axial force on oblique part of the blade outlet.

Fig. 5. Axial force components

The detailed axial force components under different conditions were calculated from the numerical simulations, as listed in Table 5. It should be noted that the axial force of the first stage is higher than that of the second stage. This occurs because the inlet of the first stage impeller is the irrotational flow, while the impeller inlet of other stages is the rotational flow. In order to calculate the axial force accurately, the single stage axial force Fs was defined as follows:


where Ffirst is the axial force of the first stage, Fsecond is the axial force of the second stage, N is the stage number. In this paper, N=5.

Table 5 Numerical simulation results of axial force


In order to further analyze the accuracy of the formula calculation results and numerical results, 150QJ20 type DCP was manufactured and tested. As shown in Fig. 6, the test-bad has the identification from the technology department in Jiangsu province of China. The pump outlet pressure is measured by a pressure transmitter with 0.1% measurement error. The volumetric flow rate of the pump was measured by a turbine flow meter, and its systematic measurement uncertainty is 0.2%. The voltage, amperes and other values of the motor were recorded during the experiments, the speed of rotation and the torque were measured by electrical measurement method, and the measurement error of the sensors is 0.2%. The overall measurement uncertainty is calculated by the square root of the sum of the squares of the systematic and random uncertainties and the calculated result of efficiency uncertainty is 0.5% (Aguirre et al., 2010; Celik et al., 2008; ISO 9006, 1999).

Fig. 6. Deep-well centrifugal pump test rig (not scale)

As shown in Fig.7, the moment sensor was installed at the bottom of the motor's rotor and connects the axial force measurement instrument, and its measurement error is 0.2%. In the test, the pump device was started in the first, after running a period of time, turn to a small flow rate and stop the pump. The value of the axial force measurement instrument was cleared to zero at this time. This ensures the measured value is the axial force of the rotating impeller. Restart the pump and record the results from the zero flow rate.

Fig. 7. Axial force measurement instrument

The detailed experimental results were shown in Table 6, and Fig. 8 indicated the comparison of axial force between experiment results and numerical results. It could be noted that the axial force was decreasing with the increase of flow rate. At the small flow rate, the high head means the pressure difference between the front shroud and rear shroud is large, which is the main

Table 6 Experimental results

Fig. 8. Comparison of the numerical and experiment results

components of total axial force. When the flow rate is less than 22m3/h (1.1Qd), the numerical result is less than the experiment result, but the contrary is the case while the flow rate higher than 22m3/h. However, for all the studied flow rates, the numerical results are consistent with the changing trend of the experiment data. Besides, at the rate flow point, the deviation of the axial force obtained by numerical simulation with the experimental value is approximately 3.8%, which confirmed the feasibility of predicting the axial force of DCP by the numerical method.

Compares the axial force experiment results with the three different calculation formulas (shown in Table 2), the result of Lu's calculation formula is most close to the experiment value. This proved that revised static pressure distribution is more closed to the actual flow field. The result of Guan (1995) formula is higher than the experiment result, the reason maybe is he assumed the static pressure on impeller outlet and pressure on the front shroud is equal. The result of Pfleiderrer (1964) formula is far less than experiment value, because his assumption didn't consider the continuity of pressure variation on impeller outlet.


In this study, the axial force in DCP was researched by using the methods of theoretical analysis, numerical simulation and experiment verification. Three different calculation formulas of axial force were introduced and compared with the experimental value. The comparison indicated that Lu's formula has relatively accurate results, which has the reasonable assumption, i.e. the liquid circumferential velocity in the lateral of impeller shroud is 0.45ω, and the liquid circumferential velocity in the oblique side of impeller outlet is 0.65ω. Furthermore, the calculated result of Lu's formula at design flow rate only less than the experimental result of 2.6%. This formula could be used for estimating the axial force during design stage, as it has the high accuracy and simple calculation process.

Three dimensional CFD numerical simulation of a two-stage DCP had been carried out at different flow conditions. The numerical results of axial force have been compared with the experimental results and found to be in good agreement with each other. At rated flow point, the deviation of the axial force obtained by numerical simulation with the experimental value is approximately 3.8%. This is a unique advantage of numerical simulation, in which one could visualize the flow field and estimate the pump performance (including the axial force) during the design stage without building the physical prototype. Besides, both the numerical results and experimental results reveal that the axial force of DCP was decreasing with the increase of flow rate, also means the DCP has a comparatively large axial force at small flow rate. As such, the DCP should avoid working long hours at smaller flow rate.




deep-well centrifugal pump


flow rate


rated flow


single-stage head


stage number


rotation speed


specific speed


blade number


blade outlet width


blade inlet angle


blade outlet angle


radius of impeller front shroud


radius of impeller middle streamline outlet


radius of seal ring


radius of impeller inlet


radius of hub


radius of impeller rear shroud


angle between impeller exit axial velocity and axis


total axial force


dynamic reaction force


potential head in impeller outlet


impeller angular velocity


pump theoretical flow rate


radius of hub


radius of seal ring


radius of impeller front shroud


radius of impeller rear shroud


angle between impeller exit axial velocity and axis


axial velocity slightly before impeller blade inlet


axial velocity slightly later impeller blade outlet


non-dimensional distance from the wall


1 impeller inlet

2 impeller outlet

h hub

m seal ring

d design point

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Received: January 31, 2012
Accepted: May 2, 2013
Recommended by Subject Editor: Adrian Lew

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