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Abstract

Keywords

Centrifugal pump amplification coefficient hydraulic turbine conversion relation

Introduction

Since pumps as turbines (PATs) are used in micro-hydro and energy recovery, they have a history of many years of research.^1–3 When pumps are used as turbines, the pump outlet goes into the hydraulic turbine inlet, and the pump inlet goes into the hydraulic turbine outlet. The pump impeller rotates in reverse under the action of high-pressure liquid. Part of the energy of the liquid is converted into mechanical energy of the hydraulic turbine through the PAT, thus realizing the aim of energy recovery.^4–7 The PAT is able to recycle energy very efficiently, but the main questions for researchers are how to choose a centrifugal pump to use as a turbine and how to make sure it has good performance and high efficiency for any set of hydraulic parameters (such as head H and flow Q).

At present, the methods of theoretical derivation and experimental regression are mainly adopted to research conversion relations of PAT, and the main conversion relations obtained by theoretical derivation are the Childs relation, the Hancock relation, the Stepanoff relation, the Sharma relation, the Alatorre-Frenk and Thomas relation, the Schmiedl relation, the Grover relation, and the Hergt relation (Appendix 2).⁸ Williams verified the above conversion relations using experimental data from 35 pumps, and he found that they are unfaithful when they are used to predict the performance of hydraulic turbines; there are large errors. Scholars also obtained relations through experimental regression, mainly including the H Nautiyal et al.⁸ relation; the conversion relations are obtained from the previous works.^9–12

Currently, these PAT conversion relations differ not only in form but also in conversion result; the calculation error is also greater. Applying on actual engineering project, the performance of selected hydraulic turbine is often not optimal because of the error of these PAT conversion relations is major. Therefore, it is very important to find out why the calculation results are different for various conversion relations and to eliminate the effect of these factors on the conversion relation of PAT. Research results will provide important theoretical support for the model selection of hydraulic turbines.

Putting forward a calculation method

Most centrifugal pumps with low specific speed are designed using the method of increasing the flow rate; by this, each designer selects different flow rate amplification coefficients for the pumps. The different amplification coefficients (such as flow rate amplification coefficient, head amplification coefficient, and specific speed amplification coefficient) lead to differences between the optimum conditions after amplification and the optimum design conditions;^13–15 thus, these factors lead to differences in the conversion coefficients of centrifugal pumps as hydraulic turbines. Therefore, the amplification coefficient is one of the main reasons that calculation results of PAT conversion relations differ between scholars. In order to eliminate the effect of the amplification coefficient on the conversion relation of centrifugal pumps as hydraulic turbines, and to choose more precisely which centrifugal pump to use as a hydraulic turbine, in this article, we use the centrifugal pump impeller entrance and outlet velocity triangles, the hydraulic turbine impeller entrance and outlet velocity triangles, the basic energy equation of centrifugal pumps, and the basic energy equation of hydraulic turbines to carry out more in-depth theoretical derivations to consider the amplification coefficient for the conversion relation of centrifugal pumps as hydraulic turbines.

Conversion relation of centrifugal pumps as hydraulic turbines

The impeller entrance and outlet velocity triangles of a centrifugal pump and hydraulic turbine are shown (Figure 1).

Figure 1.

Impeller entrance and outlet velocity triangles of a centrifugal pump and hydraulic turbine.

Assuming that $H_{t} = k_{1} H'_{p}$ , $Q_{t} = k_{2} Q'_{p}$ , and $n_{st} = k_{3} n'_{sp}$ , where $H'_{p} = ε_{1} H_{p}$ , $Q'_{p} = ε_{2} Q_{p}$ , $n'_{sp} = ε_{3} n_{sp}$ , and $ε_{1}$ , $ε_{2}$ , and $ε_{3}$ are the coefficients of head amplification, flow rate amplification, and specific speed amplification when the centrifugal pump is designed using the method of increasing the flow rate, respectively; they can be obtained using methods in the literature.¹³ $H_{p}$ , $Q_{p}$ , and $n_{sp}$ are the design head, the design flow rate, and the design specific speed of the centrifugal pump, respectively. $H'_{p}$ , $Q'_{p}$ , and $n'_{sp}$ are the head, the flow rate, and the specific speed of the centrifugal pump after increasing the flow rate, respectively. $H_{t}$ , $Q_{t}$ , and $n_{st}$ are the head, the flow rate, and the specific speed of the hydraulic turbine, respectively. $k_{1}$ , $k_{2}$ , and $k_{3}$ are the coefficients of the head conversion, the flow rate conversion, and the specific speed conversion of the centrifugal pump as a hydraulic turbine, respectively, which are used below.

Assuming that the rotation speed of the hydraulic turbine is equal to the rotation speed of the centrifugal pump under the optimum operating condition, namely, $n_{t} = n_{p}$ , then, following the specific speed equation

n_{s t} = \frac{n_{t} \sqrt{Q_{t}}}{H_{t}^{3 / 4}} = \frac{n_{p} \sqrt{k_{2} {Q^{'}}_{p}}}{{(k_{1} {H^{'}}_{p})}^{3 / 4}} = {n^{'}}_{s p} \cdot \frac{k_{2}^{1 / 2}}{k_{1}^{3 / 4}}

Therefore

k_{3} = \frac{n_{s t}}{{n^{'}}_{s p}} = \frac{k_{2}^{1 / 2}}{k_{1}^{3 / 4}}

(1)

The basic equation for the pump operating condition is

H'_{p} = \frac{η_{hp}}{g} {(u_{2} v_{u 2} - u_{1} v_{u 1})}_{p}

(2)

where $η_{hp} = η_{p} / η_{vp} η_{mp}$ , $η_{vp} = 1 / (1 + 0.68 n'_{sp}^{- 2 / 3})$ , and $η_{mp} = 1 - (0.07 / (n'_{sp} / 100 {))}^{7 / 6}$ .

The basic equation for the hydraulic turbine operating condition is

H_{t} = \frac{1}{g η_{ht}} {(u_{1} v_{u 1} - u_{2} v_{u 2})}_{t}

(3)

where $η_{ht} = η_{t} / η_{vt} η_{mt}$ , and assuming that $η_{vt} = η_{vp}$ , $η_{mt} = η_{mp}$ .

We can show that the water inlet angle to the impeller in the hydraulic turbine is equal to the volute angle. In fact, the volute operates as a guide channel. The water outlet angle from the impeller during turbine operation is equal to the impeller inlet angle during pump operation (assuming no whirl at exit).⁹ So, we can consider the same Euler heads for turbine and pump modes at best efficiency point (BEP) after increasing the flow rate⁹

{(u_{2} v_{u 2} - u_{1} v_{u 1})}_{p} = {(u_{1} v_{u 1} - u_{2} v_{u 2})}_{t}

(4)

Following equations (2), (3), and (4), then

\frac{H'_{p}}{H_{t}} = η_{hp} η_{ht}

(5)

Following equation (5)

k_{1} = \frac{1}{η_{hp} η_{ht}}

(6)

Following equation (4)

\begin{matrix} μ \cdot u_{2 p}^{2} - v_{m 2 p} \cot β_{2 p} u_{2 p} - (u_{1 p} - v_{m 1 p} \cot β_{1 p}) u_{1 p} \\ = (u_{1 t} - v_{m 1 t} \cot β_{1 t}) u_{1 t} - (u_{2 t} - v_{m 2 t} \cot β_{2 t}) u_{2 t} \\ (μ \cdot u_{2 p}^{2} - \frac{Q'_{p}}{π D'_{2 p} b'_{2 p} φ_{2 p} η_{vp}} \cot β_{2 p} u_{2 p} - u_{1 p}^{2} + \frac{Q'_{p}}{F_{1 p} φ_{1 p} η_{vp}} \cot β_{1 p} u_{1 p}) \\ = (u_{1 t} - \frac{k_{2} Q'_{p}}{π D_{1 t} b_{1 t} φ_{1 t} η_{vt}} \cot β_{1 t}) u_{1 t} - (u_{2 t} - \frac{k_{2} Q'_{p}}{F_{2 t} φ_{2 t} η_{vt}} \cot β_{2 t}) u_{2 t} \end{matrix}

(7)

where $μ = 1 - (π / z) \sin β_{2 p}$ , $φ_{2 p} = 1 - (z δ_{2} / π D'_{2 p}) \cdot \sqrt{1 + {(\cot β_{2 p} / \sin γ_{2})}^{2}}$ , $γ_{2}$ is the angle between the axial cutting line and the axial-surface streamline, ° (generally $γ_{2} = 60 \circ - 90 \circ$ ). $F_{1 p}$ is the area of the water-carrying section of the axial-surface channel in the blade inlet of the centrifugal pump, m², $F_{1 p} = 2 π Rb$ ; $R$ and $b$ are obtained by drawing the axial projection of the centrifugal impeller, $φ_{1 p} = 0.85$ . $φ_{1 t} = 1 - (z δ_{1} / π D_{1 t}) \cdot \sqrt{1 + {(\cot β_{1 t} / \sin γ_{1})}^{2}}$ , where $γ_{1}$ is the angle between the axial cutting line and the axial-surface streamline, ° (generally $γ_{1} = 60 \circ - 90 \circ$ ); $F_{2 t}$ is the area of the water-carrying section of the axial-surface channel in the blade outlet of the hydraulic turbine, m²; $F_{2 t} = 2 π Rb$ and $φ_{2 t} = 0.85$ , which are the same below.

When the centrifugal pump is used as a hydraulic turbine

\begin{matrix} D'_{2 p} = D_{1 t}, b'_{2 p} = b_{1 t}, β_{1 p} = β_{2 t}, \\ β_{2 p} = β_{1 t}, φ_{2 p} = φ_{1 t}, \\ φ_{1 p} = φ_{2 t}, F_{1 p} = F_{2 t}, u_{2 p} = u_{1 t}, u_{1 p} = u_{2 t} \end{matrix}

Assuming that

\begin{matrix} D'_{2 p} = D_{1 t} = D_{2}, b'_{2 p} = b_{1 t} = b_{2}, \\ β_{1 p} = β_{2 t} = β_{1}, β_{2 p} = β_{1 t} = β_{2}, \\ φ_{2 p} = φ_{1 t} = φ_{2}, φ_{1 p} = φ_{2 t} = φ_{1}, \\ F_{1 p} = F_{2 t} = F_{1}, u_{2 p} = u_{1 t} = u_{2}, u_{1 p} = u_{2 t} = u_{1} \end{matrix}

(8)

Plugging equation (8) into equation (7)

\begin{matrix} (μ \cdot u_{2}^{2} - \frac{Q'_{p}}{π D_{2} b_{2} φ_{2} η_{vp}} \cot β_{2} u_{2} - u_{1}^{2} + \frac{Q'_{p}}{F_{1} φ_{1} η_{vp}} \cot β_{1} u_{1}) \\ = (u_{2} - \frac{k_{2} Q'_{p}}{π D_{2} b_{2} φ_{2} η_{vt}} \cot β_{2}) u_{2} - (u_{1} - \frac{k_{2} Q'_{p}}{F_{1} φ_{1} η_{vt}} \cot β_{1}) u_{1} \end{matrix}

(9)

Assuming that $(Q'_{p} / F_{1} φ_{1}) u_{1} \cot β_{1} = C_{1}$ and $(Q'_{p} / π D_{2} b_{2} φ_{2}) u_{2} \cot β_{2} = C_{2}$ , then, rearranging equation (9) gives

μ \cdot u_{2}^{2} - \frac{C_{2}}{η_{vp}} - u_{1}^{2} + \frac{C_{1}}{η_{vp}} = u_{2}^{2} - k_{2} \cdot \frac{C_{2}}{η_{vt}} - u_{1}^{2} + k_{2} \cdot \frac{C_{1}}{η_{vt}}

(10)

From equation (10)

k_{2} = (\frac{μ - 1}{C_{1} - C_{2}} \cdot u_{2}^{2} + \frac{1}{η_{vp}}) \cdot η_{vt}

(11)

Plugging equations (6) and (11) into equation (1)

k_{3} = η_{vt}^{1 / 2} \cdot {(μ η_{hp} η_{ht})}^{3 / 4} \cdot {(\frac{μ - 1}{C_{1} - C_{2}} \cdot u_{2}^{2} + \frac{1}{η_{vp}})}^{1 / 2}

(12)

Therefore

\frac{H_{t}}{H'_{p}} = \frac{1}{η_{hp} η_{ht}}

(13a)

\frac{Q_{t}}{Q'_{p}} = (\frac{μ - 1}{C_{1} - C_{2}} \cdot u_{2}^{2} + \frac{1}{η_{vp}}) \cdot η_{vt}

(13b)

\frac{n_{st}}{n'_{sp}} = η_{vt}^{1 / 2} \cdot {(η_{hp} η_{ht})}^{3 / 4} \cdot {(\frac{μ - 1}{C_{1} - C_{2}} \cdot u_{2}^{2} + \frac{1}{η_{vp}})}^{1 / 2}

(13c)

where $C_{1} = (Q'_{p} / F_{1} φ_{1}) u_{1} \cot β_{1}$ , $C_{2} = (Q'_{p} / π D_{2} b_{2} φ_{2}) u_{2} \cot β_{2}$ , $Q'_{p} = ε_{2} Q_{p}$ , and $n'_{sp} = ε_{3} n_{sp}$ .

Equation (13a)–(13c) constitutes the conversion relation of centrifugal pumps as hydraulic turbines on the basis of the amplification coefficient, and they are non-dimensionalize. Because the new conversion relation includes the amplification coefficient of the centrifugal pump, the conversion relation may be used to choose more accurately which centrifugal pump to use as a hydraulic turbine.

Experimental study

Experimental scheme model selection

Three laboratory models were tested in the hydraulic turbine experiment table (Figure 2). The main equipment for the hydraulic turbine experiment table is a feed pump, PAT, energy dissipation pump, torque meter, electromagnetic flowmeter, differential pressure transducer, sluice valve, control valve, bottom valve, and an electromotor. The high-pressure fluid required for the PAT’s energy recovery was supplied by a feed pump. An energy dissipation pump was installed to consume energy generated by the PAT and to regulate its rotational speed.

Figure 2.

Hydraulic turbine experiment table.

A torque meter was installed to measure energy generated by the PAT. The discharge was measured by an electromagnetic flowmeter. The PAT’s inlet and outlet pressure was measured by differential pressure transducer. The flow rate of turbine is controlled by control valve. After measuring all parameters, the required pressure head, shaft power, and efficiency were obtained. The uncertainties of the measured required pressure head H, flow rate Q, hydraulic power P_h, generated shaft power P, and efficiency $η$ are $\pm 0.16 %$ , $\pm 0.50 %$ , $\pm 0.54$ , $\pm 1.20 %$ , and $\pm 1.23$ , respectively.

Centrifugal pumps with specific speeds of 33, 47, and 66 were chosen as the hydraulic turbines in this experiment. The design parameters of these centrifugal pumps are flow rate $Q_{P} = 25 m^{3} / h$ , head $H_{P} = 32 m$ , efficiency $η_{p} = 52 %$ , $η_{t} = 50 %$ , rotation speed n = 1450 r/min, and specific speed $n_{sp} = 33$ ; flow rate $Q_{P} = 25 m^{3} / h$ , head $H_{P} = 20 m$ , efficiency $η_{p} = 60 %$ , $η_{t} = 58.5 %$ , rotation speed n = 1450 r/min, and specific speed $n_{sp} = 47$ ; flow rate $Q_{P} = 25 m^{3} / h$ , head $H_{P} = 12.5 m$ , efficiency $η_{p} = 65 %$ , $η_{t} = 63 %$ ,rotation speed n = 1450 r/min, and specific speed $n_{sp} = 66$ , respectively. The geometric parameters are listed (Table 1). Figure 3 shows their impellers.

Table 1.

Geometric parameters of the centrifugal pumps.

Parts	Parameter	$n_{sp} = 33$	$n_{sp} = 47$	$n_{sp} = 66$
Impeller	Impeller inlet diameter $D_{1 p}$ /mm	48	80	80
	Inlet blade angle $β_{1 p}$ /(∘)	42	40	38
	Blade number z	5	5	6
	Impeller outlet diameter $D_{2 p}$ /mm	200	252	210
	Impeller outlet width $b_{2 p}$ /mm	6	6.5	9
	Outlet blade angle $β_{2 p}$ /(∘)	38	39	27
	Blade thickness $δ$ /mm	5	4	4
Volute	Volute base circle diameter $D_{4 p}$ /mm	205	260	215
	Volute inlet width $b_{3 p}$ /mm	12	22	26
	Volute outlet diameter $D_{5 p}$ /mm	50	50	50
	Volute sectional shape	Circular	Circular	Circular

Figure 3.

The centrifugal pumps impellers.

Experimental verification

Three centrifugal pumps are used as experimental models in Table 1, and they are allowed to rotate in reverse to act as hydraulic turbines. The new conversion coefficients of the centrifugal pumps as hydraulic turbines containing flow rate amplification coefficients are calculated by substituting the geometric parameters of these centrifugal pumps into equations (6), (11), and (12). After this, we perform an experiment in order to verify the calculation results of the conversion relations containing flow rate amplification coefficients using the hydraulic turbine experiment table shown in Figure 2.

The comparative results of the new conversion coefficients of the centrifugal pumps as hydraulic turbines are obtained using equations (6), (11), and (12); experimental data; the Childs relation; the Hancock relation; the Stepanoff relation; the Sharma relation; the Alatorre-Frenk and Thomas relation; the Schmiedl relation; the Hergt relation; the literature⁹ and are listed in Table 2.

Table 2.

Comparison between the calculation results of different conversion relations and experimental data.

Centrifugal pump	Parameters	New relation	Childs	Hancock	Stepanoff	Sharma	Alatorre-Frenk	Schmiedl	Hergt	Literature⁹	Experimental data	Error of new relation relative to experimental data (%)
$n_{sp} = 33$	$k_{1}$	2.33	1.92	2.00	2.24	2.19	2.40	3.87	1.02	2.46	2.30	+1.30
	$k_{2}$	1.97	1.92	2.00	2.24	1.69	2.00	2.34	1.22	1.72	1.91	+3.14
	$k_{3}$	0.75	−	−	0.67	−	−	−	−	−	0.73	+2.74
$n_{sp} = 47$	$k_{1}$	1.98	1.67	1.71	1.93	1.85	2.22	3.13	1.13	2.10	1.94	+2.06
	$k_{2}$	1.86	1.67	1.71	1.93	1.50	2.04	2.07	1.25	1.59	1.78	+4.49
	$k_{3}$	1.82	−	−	0.72	−	−	−	−	−	0.78	+5.13
$n_{sp} = 66$	$k_{1}$	1.89	1.54	1.59	1.83	1.68	2.07	2.90	1.18	1.93	1.86	+1.61
	$k_{2}$	1.72	1.54	1.59	1.83	1.41	2.03	1.98	1.27	1.52	1.66	+3.61
	$k_{3}$	0.81	−	−	0.74	−	−	−	−	−	0.80	+1.25

New conversion coefficients of the centrifugal pumps as hydraulic turbines are obtained using equations (6), (11), and (12) and are closer to the experimental data than the calculation results using other conversion relations (Table 2); the errors are less than those of the experimental data. The main reason for the error is the assumptions quoted in this article which are obtained from the work by Yang et al.,⁹ and in addition, the rotation speed of hydraulic turbine is also the cause of error because of the rotation speed of hydraulic turbine is hard to control. Measuring precision of flow rate and pressure is also one of the reasons.

Clearly, we can use the new conversion relation of centrifugal pumps as hydraulic turbines containing the flow rate amplification coefficient to accurately choose which centrifugal pump to use as a hydraulic turbine.

Effect of the amplification coefficient on the conversion coefficient

We know that the conversion relation of centrifugal pumps as hydraulic turbines is indeed connected to the flow rate amplification coefficient and the specific speed amplification coefficient from equation (13a)–(13c). The conversion coefficients when the specific speed is equal to 33 are listed for different amplification coefficients (Table 3).

Table 3.

Conversion coefficients under different amplification coefficients.

Serial number	Flow rate amplification coefficient $ε_{2}$	Head conversion coefficient $k_{1}$	Flow rate conversion coefficient $k_{2}$	Specific speed conversion coefficient $k_{3}$
1	1.44	2.33	2.08	0.77
2	1.48	2.33	2.05	0.76
3	1.52	2.33	2.02	0.75
4	1.56	2.33	2.00	0.75
5	1.60	2.33	1.97	0.75
6	1.64	2.33	1.95	0.74

Table 3 shows that with an increase in flow rate amplification coefficient, the flow rate conversion coefficient and specific speed conversion coefficient gradually decrease; the effect of flow rate amplification coefficient on the flow rate conversion coefficient is greater; the flow rate amplification coefficient has no effect on the head conversion coefficient. It is clear that the effect of the different amplification coefficients on the conversion coefficient of centrifugal pumps as hydraulic turbines is greater when the centrifugal pump with a low specific speed is designed with the method of increasing the flow rate.

Conclusion

In this article, we obtain a new conversion relation of centrifugal pumps as hydraulic turbines that contains the flow rate amplification coefficient. We found that with an increase in flow rate amplification coefficient, the flow rate conversion coefficient and specific speed conversion coefficient gradually decrease, and the effect of flow rate amplification coefficient on the flow rate conversion coefficient is greater; the flow rate amplification coefficient has no effect on the head conversion coefficient.

We know by research that when centrifugal pumps with low specific speed are designed using the method of increasing the flow rate, there are different flow rate amplification coefficients for the pumps. These different amplification coefficients lead to differences between the optimum conditions after amplification and the optimum design conditions. Therefore, these different flow rate amplification coefficients lead to differences in the conversion coefficients of centrifugal pumps as hydraulic turbines, and the flow rate amplification coefficient is the main reason that calculation results of PAT conversion relations differ between scholars.

Footnotes

Appendix 1

Appendix 2

The model of reference.⁸

Academic Editor: Yangmin Li

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 supported by grants from the Education Department Key Project of Sichuan Province of China (17ZA0366), the Nature Science Foundation of China (51279172), China Postdoctoral Science Foundation (2016M600090), the Key Scientific Research Fund of Xihua University of China (Z1510417), and the Open Research Subject of Key Laboratory of Fluid and Power Machinery, Ministry of Education (szjj2016-004).

References

Yang

Liu

Kong

. Experimental, numerical, and theoretical research on impeller diameter influencing centrifugal pump-as-turbine. J Energ Eng 2013; 139: 299–307.

Wang

Kong

. Researching status of centrifugal pump as turbine. J Drain Irrigat Mach Eng 2013; 31: 674–680.

Shi

Yang

Miao

. Unsteady calculation of pressure pulsations within hydraulic turbine under different entrance sections. J Aerosp Power 2016; 31: 559–668.

Shi

Yang

Wang

. Effect of entrance section of hydraulic turbine on pressure pulsations within volute. J Xihua Univ 2016; 35: 7–12.

Yang

Kong

Chen

. Effects of blade wrap angle influencing a pump as turbine. J Fluid Eng: T ASME 2012; 134: 1–10.

Yang

Kong

. Influence of blade number on the performance and pressure pulsations in a pump used as a turbine. J Fluid Eng: T ASME 2012; 134: 1–10.

Yang

Liu

Kong

. Effects of the radial gap between impeller tips and volute tongue influencing the performance and pressure pulsations of pump as turbine. J Fluid Eng: T ASME 2014; 136: 1–9.

Nautiyal

Varun

Kumar

. Experimental investigation of centrifugal pump working as turbine for small hydropower systems. Energ Sci Tech 2011; 1: 79–86.

Yang

Derakhshan

Kong

FY.

Theoretical, numerical and experimental prediction of pump as turbine performance. Renew Energ 2012; 48: 507–513.

10.

Derakhshan

Kasaeian

Optimization, numerical, and experimental study of a propeller pump as turbine. J Energ Resour: ASME 2014; 136: 1–11.

11.

Singh

Nestmann

An optimization routine on a prediction and selection model for the turbine operation of centrifugal pumps. Exp Therm Fluid Sci 2010; 34: 152–164.

12.

Yang

Huang

. Research on different specific speed pumps used as turbines. Trans Chin Soc Agric Mach 2013; 44: 69–72.

13.

Yang

Zhang

RH.

Calculation method of the increasing coefficients for low specific-speed centrifugal pump. J Mech Eng 2005; 41: 203–205.

14.

Zhao

Liu

. Effects of blade thickness on performance of low-specific-speed centrifugal pump. J Drain Irrigat Mach Eng 2015; 33: 1033–1037.

15.

Yuan

Wang

Pei

. Multi-objective optimization of low-specific-speed centrifugal pump. Trans CSAE 2015; 31: 46–52.

Conversion relation of centrifugal pumps as hydraulic turbines based on the amplification coefficient

Abstract

Keywords

Introduction

Putting forward a calculation method

Conversion relation of centrifugal pumps as hydraulic turbines

Experimental study

Experimental scheme model selection

Experimental verification

Effect of the amplification coefficient on the conversion coefficient

Conclusion

Footnotes

Appendix 1

Appendix 2

Declaration of conflicting interests

Funding

References