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Abstract

To study the effects of different guide vane numbers on the unsteady performance of pump as turbine based on the Navier–Stokes equation and standard k-epsilon turbulence model, computation fluid dynamics technology was used to simulate the flow field in pump as turbine. The turbulent kinetic energy, unsteady radial forces, and power losses were also clarified. The results show that for turbines with a guide vane, the distribution of turbulent kinetic energy is more uniform than before, and the radial force vector distribution is more symmetrical within four quadrants. The time-domain distribution of radial force is more periodic, the number of fluctuation periods is equal to the guide vane numbers, and the dominant frequency of the radial force is equal to the blade frequency. For different guide vane numbers, the effects on the unsteady performance of pump as turbine are different. When the guide vane number is equal to 9, the distribution of turbulence kinetic energy is optimal. In addition, at an optimal flow rate, both the time domains of the radial force and the power losses of the impeller are minimal, so to the geometric parameters of the hydraulic turbine are definite, the optimal guide vane number exists.

Keywords

Pump as turbine guide vane numbers turbulent kinetic energy radial force power losses

Introduction

A centrifugal pump running in inverse mode (pump as turbine) can be applied to residual pressure energy recovery in industries. To improve the efficiency and stability of pump as turbine, many theoretical inferences, numerical simulations, and experiments have been completed, and research attention has primarily focused on the relationships of the external characteristic parameters between pumps and turbines.^1–5 However, the results are not ideal and differ greatly. In recent years, studies on the optimization of turbine stator and rotor parts have been conducted by researchers, and mathematical calculation methods have been adopted to optimize the blade curves;^6–10 both splitter blades^11,12 and forward-curved blades¹³ were used in turbine impellers in most studies. It is shown that splitter blades play a positive role in inverse mode operation such that the more the splitter blade is, the more stable the flow field is, and the lower the head of the turbine, the better the efficiency. The efficiency curve of pump as turbine with forward swept blades is flatter, and the high efficiency operating range of an impeller with forward swept blades is wider than that of a traditional turbine. Furthermore, the efficiency and stability can be improved by adding guide vanes for the pump as turbine. In a pump as turbine with a guide vane, the inflow conditions were improved,^14,15 the pressure fluctuation was greatly decreased, and the vibration and noise were decreased.^16,17 For different guide vane numbers, the influences on the efficiency and stability are different; fewer studies exist on guide vanes, and studies on the influence of guide vane numbers on the distribution of turbulence kinetic energy, unsteady radial force, and unsteady power losses are nonexistent. In this article, based on the verification of the accuracy of computation fluid dynamics (CFD), pump as turbine and pump with guide vane as turbine were chosen as the research objects. An unsteady three-dimensional (3D) simulation based on Reynolds time-averaged governing equations and the standard $k - ε$ turbulence model was presented, and the effects of different guide vane numbers on the distribution of turbulence kinetic energy, unsteady radial force, and unsteady power losses are obtained. Finally, some theoretical bases for the stability of pump as turbine operation are provided.

Research objects and geometric modeling

In Yang et al.¹⁵ a pump as turbine was chosen as the research object. At the best efficiency point, the design parameters of pump as turbine are as follows: flow rate $Q = 170 m^{3} / h$ , head $H = 32 m$ , speed $n = 1450 r / min$ , and specific speed $n_{s} = 84.5$ . A different guide vane was added to the pump as turbine, and four models were established. The guide vane numbers are Z₀ = 0, Z₀ = 7, Z₀ = 9, and Z₀ = 11. In this article, the four models are adopted, and the main geometric parameters of the original turbine and remodeled turbines are shown in Table 1. The 3D pro/E modeling is shown in Figure 1.

Table 1.

Main geometric parameters of the original and remodeled turbines.

	Impeller						Volute			Guide vane
	$D_{1}$	$β_{2}$	$β_{1}$	$Z$	$D_{2}$	$b_{2}$	$D_{4}$	$b_{3}$	$D_{5}$	$α_{a}$	$α_{d}$	$Z_{0}$	$b_{0}$
Original turbine	132	29.5	36	6	328	18	340	44	100	–
Remodeled turbine	132	29.5	36	6	328	18	456	40	113	17	14	7, 9, 11	18

Figure 1.

Three-dimensional modeling for pump as turbine with different guide vane numbers: (a) Z₀ = 0, (b) Z₀ = 7, (c) Z₀ = 9, and (d) Z₀ = 11.

Calculation method

Unstructured grids are meshed in the 3D pro/E models of pump as turbine, and the grid independence is verified under the steady state. When the grid numbers of pump as turbine with guide vane are more than $1.2 \times 10^{6}$ , the change of efficiency is less than 0.3%, and the final grid number of Z₀ = 0 and Z₀ = 11 is 1,031,382 and 1,700,732, respectively. The efficiency curve variation with mesh number in Z₀ = 9 is shown in Figure 2. Four parts are involved in the computational domain as follows: volute, guide vane, impeller, and draft tube. To make a full fluid flow development in the outlet section, the draft tube was extended appropriately.

Figure 2.

Variation of efficiency with mesh number in Z₀ = 9.

Based on the Reynolds time-averaged equations and the standard $k - ε$ model, an unsteady 3D simulation was presented for pump as turbine. In the process of unsteady numerical simulation, multiple coordinate systems were adopted. The flow field of the impeller was calculated in the rotating coordinate system, and the flow fields of the volute and guide vane were calculated in the static coordinate system. In the steady and unsteady calculation, the frozen rotor interface and the transient rotor/stator interface were used for dynamic and static coupling interfaces, respectively. The velocity condition was used for the inlet boundary condition, and the pressure condition was used for the outlet boundary condition. The solid walls of the hydraulic model included the volute and guide vane, in which no slip boundary condition was adopted. The calculation accuracy of convergence is 10⁻⁵, and a complete calculation was done from the volute inlet to the draft tube outlet. The unsteady calculation is based on the steady calculation convergence. The time step of the unsteady numerical simulation is 3.448 × 10⁻⁴ s, and the angle of rotation of the impeller for each cycle is 360°; therefore, the impeller rotates once every 120 time steps. The impeller rotated a total of six cycles during the unsteady calculation process, and the data of the sixth cycle were adopted for analysis.

Results and analysis

Test

The test bench for pump as turbine is shown in Figure 3. The accuracy of the numerical calculation is verified by the experiment.

Figure 3.

Test bench for pump as turbine.

The comparison between experimental and numerical performance curves for models Z₀ = 0 and Z₀ = 9 are listed in Figure 4. From the Figure 4, the performance curves of numerical prediction agree well with that of the experimental results. The efficiency of the numerical prediction is higher than that of experimental values because the numerical calculation considers the leakage loss by balancing the holes and mechanical losses when not considering the mechanical seals and bearings. At the best efficiency point, in the model of Z₀ = 0, the relative errors of the efficiency and head are 3.99% and 3.19%, respectively; in the model of Z₀ = 9, the relative error of the efficiency and head are 2.34% and 4.2%, respectively. It is fully proven that the turbulence model in the description of fully developed turbulent flow has good applicability, so the performance of pump as turbine can be better predicted by the numerical method. The numerical method can then be used to calculate the unsteady characteristic of pump as turbine.

Figure 4.

Comparison between the test and numerical performance curves.

Analysis of turbulence kinetic energy under variable working conditions

The asymmetry of the volute and rotor–stator interaction causes turbulence kinetic energy concentration to appear in pump as turbine. Theoretically, the turbulence kinetic energy equals half of the product of turbulent velocity fluctuation variance and fluid mass, which generally is expressed by the physical quantity $k$ and can be calculated through the turbulence intensity $I$ . The turbulence intensity $I$ is used to judge the flow conditions in pump as turbine; the greater the turbulence intensity is, the more unstable the flow state and the greater the flow losses. The formulas of turbulence kinetic energy and turbulence intensity $I$ are as follows¹⁸

k = \frac{3}{2} (uI)^{2}

(1)

I = 0.16 {R_{e}}^{\frac{- 1}{8}}

(2)

where $u$ is the average velocity, $I$ is the intensity of turbulence, and $R_{e}$ represents the Reynolds number.

In the middle section of the turbine, the contour of turbulence kinetic energy is as shown in Figure 5. From Figure 5, the kinetic energy in the volute is more uniform, the centralized kinetic energy appeared in the impeller, the volute tongue, the dynamic and static coupling interface, and so on. In the dynamic and static coupling interface, the kinetic energy is the maximum, which is caused by the rotor–stator interaction. In the model of Z₀ = 0 and in the dynamic and static coupling interface, the centralized kinetic energy is obvious. The higher the flow rate is, the more obvious the concentration of kinetic energy, so at a low flow rate, the kinetic energy is less. The centralized area is also less and not obvious, but at a high flow rate, the kinetic energy is greater, while the centralized area is also greater and more obvious. At the flow rate of 0.8Q, only a small amount of kinetic energy centralized near the volute tongue and the suction surface of the impeller. At a flow rate of 1.2Q, the kinetic energy centralized in the pressure surface of the impeller; however, at a flow rate of 1.4Q, the kinetic energy centralized both in the suction surface and pressure surface of the impeller.

Figure 5.

Contour of turbulence kinetic energy in turbine middle section: (a) Z₀ = 0, (b) Z₀ = 7, (c) Z₀ = 9, and (d) Z₀ = 11.

It is shown that the guide vane numbers have great effects on the kinetic energy. When the guide vane numbers are different, the distributions of kinetic energy in pump as turbine are also different. In Figure 5(b)–(d), the distribution of kinetic energy in volute and guide vane is very uniform. The kinetic energy in volute tongue vanished, but the kinetic energy only appeared in the dynamic and static coupling interface of the guide vane and the impeller and the pressure surface of the impeller. At a flow rate of 0.8Q, the distribution of kinetic energy is not uniform, which at 1.4Q, it is the second, and at 1.2Q, it is the most uniform. Under different working conditions, the kinetic energy in the model of Z₀ = 7 is more condensed. The model of Z₀ = 11 is second, while the distribution of kinetic energy in the model of Z₀ = 9 is optimal; it is shown that in pump as turbine, the flow status can be changed by the different guide vane numbers, by which the kinetic energy can be distributed more uniformly. The operation of the turbine is more stable, so the optimal guide vane number exists.

Analysis of radial force vector on impeller under variable working conditions

The pump as turbine is different from pump, the radial force on the impeller exists under each working condition, and the radial force is composed of pressure and viscosity forces. In a rectangular coordinate system, the radial force can be projected in the x and y directions, so the radial force vectors F_X and F_Y can be obtained. The radial force vector diagrams on a turbine impeller under different working conditions are shown in Figure 6, and the effect of different guide vane numbers on radial force performance was analyzed. In the coordinate system, the z-axis was selected as a rotating axis, and the radial force components were F_X and F_Y. Figure 6 shows that in one cycle, the size and direction of radial force change at every moment, and the flow rate increases the size of the radial force, which also increases drastically. In the model of Z₀ = 0, the distribution of the radial force is oval. Under different working conditions, the distribution of radial force center tends to the first and fourth quadrant, and the lower the flow rate is, the more serious the distribution. In the models of Z₀ = 7, Z₀ = 9, and Z₀ = 11, the distribution of radial force is a star, and the radial force distributes periodically; the period number equals the guide vane number. At a low flow rate, the periodic fluctuation is greater; at a high flow rate, the periodic fluctuation is lesser. Under different working conditions, the distribution of radial force in four quadrants is more uniform, especially under the flow rate of 1.2Q and 1.4Q. The radial force in the model of Z₀ = 9 is the smallest, the distribution of radial force is the most uniform. At the flow rate of 1.2Q, the distortion of radial force mainly closes to the ellipse, and the periodic fluctuation is the lightest.

Figure 6.

Vector diagrams of radial force on impeller under variable working conditions: (a) 0.8Q, (b) 1.2Q, and (c) 1.4Q.

Analysis of time domain of radial force on impeller under variable working conditions

Under different working conditions, the time-domain spectrums of radial force on the impeller are shown in Figure 7. The abscissa represents the time in one cycle (the sixth rotating cycle is 0.2069–0.2483), and the ordinate represents the resultant force of transient radial force. The maximum difference of radial resultant force is defined as follow

F^{'} = F - Δ F

(3)

where $F^{'}$ is the maximum difference of radial force, $F$ is the resultant force of transient radial force, and $Δ F$ is the average radial force. The maximum differences of radial force under different guide vane numbers and different working conditions are shown in Table 2. From Figure 7 and Table 2, at the flow rate of 0.8Q, the radial force is less, but at the flow rate of 1.4Q, the radial force is greater. In the model of Z₀ = 0, under different working conditions, the distribution of the resultant force is not uniform, and the period is weak. At a time of t = 0.23 s, greater waves appeared, and the maximum radial force difference is 69.82, 82.13, and 111.39 N. The greater the flow rate is, the greater the maximum radial force difference. In the model of Z₀ = 7, at the flow rate of 0.8Q, a greater radial force wave appeared, and the maximum radial force difference is 179.22 N. At the flow rate of 1.2Q, the radial force in the model of Z₀ = 9 is the minimum, and the maximum difference is also the minimum. Therefore, the radial force on the impeller can be reduced by the guide vane; at the same time, the turbine of other geometric parameters is definite, and the optimal guide vane number exists to make the radial force on the impeller is minimal.

Figure 7.

Time-domain spectra of radial forces on impeller under different working conditions: (a) 0.8Q, (b) 1.2Q, and (c) 1.4Q.

Table 2.

Maximum radial force difference in different guide vane numbers and under variable working conditions (N).

Guide vane numbers	Z₀ = 0	Z₀ = 7	Z₀ = 9	Z₀ = 11
0.8Q	69.82	179.22	79.45	112.93
1.2Q	82.13	137.76	64.03	99.96
1.4Q	111.39	130.60	74.02	92.30

Analysis of frequency domain of radial force on impeller under variable working conditions

Under variable working conditions, the frequency-domain spectrums of radial force on impellers are shown in Figure 8, which are transformed from Figure 7 through fast Fourier transform (FFT). The dominant frequency amplitudes of radial force under variable working conditions are shown in Table 3. The rotating speed of pump as turbine is $n = 1450 r / min$ , the impeller blade number is equal to 6, so the rotational frequency is 24.17 Hz, and the blade-passing frequency is 145.02 Hz. From Figure 8 and Table 3, in the model of Z₀ = 0, the dominant frequency equals the rotational frequency, but in the model of Z₀ = 7, Z₀ = 9, and Z₀ = 11, the dominant frequency equals the blade frequency. At the low flow rate, the dominant frequency amplitude of Z₀ = 7 is the maximum, but at the optimal and high flow rate, the dominant frequency amplitude of Z₀ = 0 is the maximum. At the optimal flow rate, the dominant frequency amplitude in the model of Z₀ = 0 is 1.09 times that in the model of Z₀ = 7, 2.71 times that in the model of Z₀ = 9, and 1.56 times that in the model of Z₀ = 11. Among all the flow rates, the dominant frequency amplitude in the model of Z₀ = 9 is the minimum; the high frequency fluctuation is also the least, so the vibration and noise on the impeller can be reduced by the appropriate guide vane.

Figure 8.

Frequency-domain spectra of radial forces on the impeller under different working conditions: (a) 0.8Q, (b) 1.2Q, and (c) 1.4Q.

Table 3.

Dominant frequencies of radial force under variable working conditions (N).

Guide vane numbers	Z₀ = 0	Z₀ = 7	Z₀ = 9	Z₀ = 11
0.8Q	27.23	37.45	15.14	23.46
1.2Q	29.98	26.06	11.04	19.28
1.4Q	36.14	25.69	11.64	26.02

Analysis of power losses on impeller under variable working conditions

During the process of energy recycling, the pressure energies were converted into mechanical energies through the pump as turbine; then, the pump or fans were driven by these mechanical energies. For the pressure energies going through the impeller, only some of them were converted into mechanical energies; the rest were lost. According to the law of conservation and transformation of energy, the power loss equals the difference in total pressure energy and network; the formula is as follows

Δ P = Q (p_{1} - p_{2}) - M ω

(4)

where $Δ P$ is the power loss, $Q$ is the flow rate, $p_{1} - p_{2}$ is the pressure difference of impeller inlet and outlet, $M$ is the torque, and $ω$ is the rotating angular velocity of the impeller. After the pump as turbine added a guide vane, the power losses on the impeller were also changed, so it is necessary to analyze the influence of guide vane numbers on the impeller. The time-domain diagrams of power losses on the impeller under different guide vane numbers and different working conditions are shown in Figure 9. The abscissa represents time in one cycle (the sixth rotating cycle is 0.2069–0.2483), and the ordinate represents power losses. It is shown that in one cycle, the power losses of the turbine impeller change periodically. In the model of Z₀ = 0, with the increases in the flow rate, the power loss first decreases then gradually increases. In the model of Z₀ = 7, Z₀ = 9, and Z₀ = 11, the power losses first increase then decrease gradually. Thus, at the flow rate of 1.0Q and 1.2Q, the power loss of Z₀ = 0 is small. In Figure 9(a)–(c), the power losses of Z₀ = 0 are less than those of Z₀ = 7, Z₀ = 9, and Z₀ = 11. The lower the flow rate is, the larger the power loss differences. In Figure 9(d), the power loss differences are the minimum, and in Figure 9(e), the power loss of Z₀ = 0 is more than those of Z₀ = 7, Z₀ = 9, and Z₀ = 11. Therefore, the relationship between power loss difference with flow rate first decreases then gradually increases. In the model of Z₀ = 0, the flow rate of the optimal condition is less than those of other three models; with the increases in the flow rate, the turbine with guide vane gradually comes into the optimal condition. Therefore, for the same working conditions, the low flow rate is good for the turbine without guide vane, while the high flow rate is good for the turbine with guide vane. The average power losses in the turbine impeller are shown in Table 4. From Table 4, under the off-design condition, the power losses on the turbine impeller are larger but are smaller at the optimal condition. For the turbine with guide vane, the power loss in Z₀ = 7 is the maximum, and the power loss in Z₀ = 11 is the minimum. In the model of Z₀ = 0, the minimal power loss appeared at the flow rate of 1.0Q, but in the model of Z₀ = 7, Z₀ = 9, and Z₀ = 11, the minimal power loss appeared at the flow rate of 1.2Q. At the flow rate of 1.2Q, the average power loss in the model of Z₀ = 7 is 1.54 times that in the model of Z₀ = 0, 1.05 times that in the model of Z₀ = 9, and 1.12 times that in the model of Z₀ = 11. Thus, after the pump as turbine added a guide vane, the power losses of the turbine at a low flow rate are increased and are decreased at a high flow rate. This is also the reason that the high efficiency areas of turbine with guide vane are found at the high flow rate area.

Figure 9.

Time-domain diagrams of power losses on the turbine impeller: (a) 0.8Q, (b) 1.0Q, (c) 1.2Q, (d) 1.4Q, and (e) 1.6Q.

Table 4.

Average power losses on turbine impellers under different guide vane numbers (W).

Flow rate	0.8Q	1.0Q	1.2Q	1.4Q	1.6Q
Z₀ = 0	2889.06	1898.86	1921.71	3178.25	5928.92
Z₀ = 7	7368.52	3963.90	2955.52	3107.77	4375.60
Z₀ = 9	6586.38	5312.10	2819.00	2939.01	4256.44
Z₀ = 11	7128.57	3393.40	2634.25	2845.92	4263.51

Conclusion

The flow status can be improved by adding guide vanes for pump as turbine. The distribution of turbulence kinetic energy is more uniform than before, and the operation of pump as turbine is steadier. In the pump as turbine without guide vane, the center of radial force vector tends toward the first and fourth quadrants. The time-domain distribution of radial force is not homogeneous, the dominant frequency of radial force equals the rotating frequency, the power loss at the low flow rate is less, and at the high flow rate, it is greater. In the pump as turbine with guide vane, the radial force on impeller decreases, the distribution of radial force vector is more symmetrical within four quadrants, the time domain of radial force is more periodic, and the numbers of cycle equal the guide vane numbers. The dominant frequency of the radial force equals the blade frequency, the power losses at a low flow rate are greater, and the losses are less at a high flow rate. Generally, at a high flow rate, the energy recovery performance of pump as turbine is better, so adding a guide vane for the pump as turbine is helpful to reduce the power losses at a high flow rate.

When the guide vane number is equal to 9, the distribution of turbulence kinetic energy is optimal, the distribution of radial force is the most uniform, the time domain of radial force is at a minimum, and the power losses are also smaller. To a pump as turbine of other geometric parameters are definite, the optimal guide vane number exists.

Footnotes

Appendix 1

Handling Editor: Assunta Andreozzi

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was financially supported by the National Natural Science Fund Project of China (grant no. 51569013).

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Analysis on the effect of variable guide vane numbers on the performance of pump as turbine

Abstract

Keywords

Introduction

Research objects and geometric modeling

Calculation method

Results and analysis

Test

Analysis of turbulence kinetic energy under variable working conditions

Analysis of radial force vector on impeller under variable working conditions

Analysis of time domain of radial force on impeller under variable working conditions

Analysis of frequency domain of radial force on impeller under variable working conditions

Analysis of power losses on impeller under variable working conditions

Conclusion

Footnotes

Appendix 1

Declaration of conflicting interests

Funding

References