Optimization of large-scale wind-powered centrifugal pumps

Abstract

A study on the optimal matching of centrifugal pumps and large-scale wind turbines is reported. The study includes mathematical modelling of the performance of high-capacity centrifugal pumps when matched with efficient high-speed low-solidity wind turbines. The dependence of the performance of wind pumps on the dimensionless parameter v/√(gh), termed the wind pump’s Froude number, was fully analysed. Similar to earlier reported results on wind-powered piston pumps, analysis is showing that the wind pump’s Froude number is the dominant independent dimensionless parameter that fully defines the system’s performance. Operation with continuous variable transmission was analysed and shown to optimize the wind pump’s performance, particularly at medium and low wind speeds. The optimum relation between the required gearbox transmission ratio and Froude number of a wind-powered centrifugal pump was analytically established for three different pumps. In addition to continuous variable transmission operation, the analysis helps in optimally sizing constant transmission ratio systems.

Keywords

Wind pumping centrifugal pumps Froude number continuous variable transmission pumped storage

Introduction

Upgrading of wind pumping technologies is seen as a logical step contributing to the alleviation of shortage in water supply in arid regions. It is reported that ‘the agri-food sector alone accounts for 30% of global energy use’ (Global Status Report REN21 Renewables, 2015). The unfulfilled water need, that goes largely unmeasured, is probably much greater.

Regarding water pumps, use of rotodynamic pumps is by far the most widely used. This is due to its limited system size, limited maintenance and good performance for large capacity applications. The working efficiency of a centrifugal pump requires a high rotational speed, in contrast to a piston pump where low speed is a feature. Thus, similarly to wind generators, centrifugal pumps are coupled to high-speed low-solidity wind turbines.

A major advantage for such a match between centrifugal pumps and low-solidity turbines is the fact that the latter are much more efficient than their low-speed high-solidity counterparts. Advantages of centrifugal pumps also include the absence of pulsation in flow lines and a much smaller susceptibility to abrasion. Moreover, due to the constant torque-speed characteristics of the piston wind pump, the net conversion factor is relatively low at operating conditions not close to that of the system design condition (Bergey, 1998; Sathyajith, 2006).

Practical experience with wind-driven centrifugal pumps, largely in the United States, moved gradually towards electrically powered pumps over those mechanically coupled to the wind turbines. The main reason for such a trend is that the best wind resource points and water wells are not necessarily close and that deems mechanical transmission impractical.

The literature on centrifugal pumps is extensive. A comprehensive and authoritative view on the efficiency levels of commercial centrifugal pumps can be drawn from Karassik et al. (2001). The main parameters to be considered for high efficiency choice of centrifugal pumps are the specific-speed N_s and the capacity at the best operating point. The higher the specific-speed and capacity the higher is the pump efficiency. The reference indicates that an efficiency of 89% is a ‘representative’ value for a pump of 48 metric specific-speed value and a capacity of 2273 m³/h.

It was noted that these performance indicators improved since these quoted figures were published. Bergey (1998), for example, indicates that the efficiency range of centrifugal pumps used for pumped storage is 0.90–0.92.

Use of centrifugal pumps for wind pumping was also extensively analysed and reported. According to Sathyajith (2006) and Bergey (1998), they have a good reputation of reliability.

What remains to be upgraded is the control functions that continuously optimize the match of wind turbines and centrifugal pumps. This is the central contribution of this study. It trails from previous reported results on the continuous matching of conventional piston pumps and wind turbines (Siddig, 2010a, 2010b). The main outcome of those studies is the establishment of the dimensionless parameter, v/√(gh), as the single independent parameter that determines the wind pump’s behaviour. It shall be shown that the same is true for centrifugal pumps.

Generalized performance theory of wind-powered centrifugal pumps using continuous variable transmission

The mathematical model

The main criterion for this analysis is to control the turbine rotational speed, n_t, as to correspond continuously to the design tip-speed ratio of the wind turbine, λ_D. The tip-speed ratio λ is defined as

λ = \frac{π d n_{t}}{v}

(1)

Its design value is the one corresponding to the maximum power coefficient of the wind turbine C_PM. The extra relations needed to optimize the system are

P_{a} = \frac{1}{2} ρ_{a} A v^{3}

(2)

where P_a is the site’s wind power available to the turbine.

The pump’s hydraulic output power is then

P_{p} = g ρ_{w} h q = η C_{P M} P_{a}

(3)

where η is the net efficiency, aside from the turbine rotor power coefficient.

The step-up ratio of the gearbox is defined here as ratio of the pump speed to that of the turbine rotor

r_{n} = \frac{n_{p}}{n_{t}}

(4)

The wind pump’s Froude number, which is a form of the well-known fluid mechanics dimensionless parameter, is defined here as

F r = \frac{v}{\sqrt{g h}}

(5)

where v is the wind speed and h is the vertical water lift.

The above definition of the wind pump’s Froude number appeared first in the publication (Siddig, 1996).

Considering a wind-powered centrifugal pump of a turbine-to-water efficiency η, the net wind-to-water power conversion factor is

\frac{w a t e r p o w e r}{w i n d p o w e r} = C_{P} η = \frac{ρ_{w} g h q}{\frac{1}{2} ρ_{a} A v^{3}} = \frac{8 ρ_{w} g h q}{π ρ_{a} d^{2} v^{3}}

Optimum performance of particular wind turbine design indicates

\frac{w a t e r p o w e r}{w i n d p o w e r} = C_{P M} η = f (\frac{v}{\sqrt{g h}}, \frac{q}{d^{3} n_{p}}) 1

(6)

Noting that the wind speed v and the pump head h are the only system constraints on a known turbine-pump system, then there must exist a particular relationship tying the optimum performance, quantified as C_Pη, and Froude number Fr. The pump capacity is deducible from the power–head–capacity relationship known for pumps.

Section ‘Demonstrations based on actual turbine and pump specifications’ shall be seen to validate this approach using actual turbine and pump specifications drawn from Nelson et al. (1985), Hydraulic Institute (2016), KSB ETA pumps (n.d.) and Addison (1964).

Furthermore, for backward-bladed centrifugal pumps, the interdependence of head, rotational speed and capacity can be proved (Addison, 1964) to be expressible as

h = a N^{2} - b q^{2}

(7)

where N is the pump’s rotational speed, q is the pump’s capacity and a and b are empirical values that depend on the particular pump.

The pump efficiency dependence on rotational speed and capacity can also be specified empirically for any particular pump

η_{p} = f (q, N)

(8)

Demonstrations based on actual turbine and pump specifications

Turbine specification

To investigate the validity of the predicted dependence of the wind pump’s dimensionless operational parameters on the v/√(gh) parameter, the typical turbine characteristics of Figure 1, as compiled from Nelson et al. (1985), were used. The quoted characteristics are those of the turbine performance at a 3° pitch angle.

Figure 1.

Turbine’s power-speed characteristics, compiled from Nelson et al. (1985).

The (λ, C_P)) coordinates of the best operation point are (7.6, 0.44). These are then the design tip-speed ratio λ_D and the maximum power coefficient C_PM. The turbine’s power-speed variation was found to be presentable mathematically with the relation

C_{P} = 0.00058 λ^{4} - 0.018 λ^{3} + 0.18 λ^{2} - 0.6 λ + 0.71

(9)

Pumps specification

To check the generality of the suggested model, three pumps of different makes and different sizes were considered. Table 1 shows the basic features of the two pumps.

Table 1.

Pumps specification.

	Manufacturer	H (m)	Q (L/s)	N_Max (r/min)	Reference
I	Gambica	47	208	1480	Hydraulic Institute (2016)
II	KSB	24	7	3000	KSB ETA pumps (n.d.)
III	Addison	33	200	3000	Addison (1964)

The reason for selecting these particular makes is the availability of detailed variable speed characteristics for these models.

Sizing of the wind turbines

Case 1: the Gambica pump–based system

The first pump to connect to the turbine is the Gambica pump (Hydraulic Institute, 2016). The pump’s variable speed characteristics are shown in Figure 2.

Figure 2.

Variable speed characteristics of the Gambica pump (Hydraulic Institute, 2016).

Applying equations (7) and (8) to the specified pump leads to the expressions

h = 0.0000278 N^{2} - 326 q^{2}

(10)

\begin{matrix} η_{p} = (0.00552 - 3.8 e - 6 N) + (25.62 - 0.01066 N) q \\ + (- 179.24 + 0.09716 N) q^{2} + (350 - 0.211 N) q^{3} \end{matrix}

(11)

A check of the accuracy of the curve-fitting performed,indicated an average error of 2% for the flow rate simulation and an average error of 3% for the pump efficiency. Table 1 and equations (10) and (11) constitute the information needed to size the turbine and, consequently, analyse the turbine/pump system.

The criterion for sizing is to control the turbine such that it operates continuously at its best operation point at the effective main-stream wind speed. This latter value should normally depend on the wind distribution and the optimum energy yield. The result of the sizing is shown in Table 2.

The comments in brackets in Table 2, mechanical transmission and electrical transmission, indicate the two wind pump power transmission options, namely a pure mechanical transmission option between the turbine and the pump and the option of a remote wind generator connected via a cable to a motor-pump unit at the water source.

Table 2.

Turbine sizing results for the selected Gambica model.

$ρ_{a}$ (kg/m³)	v (m/s)	H (m)	Turbine diameter d (mechanical transmission) (m)	Turbine diameter d (electrical transmission) (m)	C_PM	$λ_{D}$
1.226	7.0	47	40.2	44.2	0.442	7.6

The reason for the larger turbine diameter in the case of the electric centrifugal pump is the compensation for the extra losses incurred due to the generator and the motor. Furthermore, it can be shown that at a reference wind speed of 7.0 m/s, and a pump head of 47 m, the design Froude number, v/√(gh), is 0.33 and the corresponding gearbox transmission ratio is 64 for the electrical transmission case.

The mechanical and electrical cable transmission efficiencies were assumed fixed at 95%. The rated generator and motor efficiencies were assumed to be 93% and 92%, respectively.

Case II: the KSB pump–based system

The second system to analyse is that involving the KSB make (KSB ETA pumps (n.d.)) as the pumping component. Its variable speed characteristics are shown in Figure 3.

Figure 3.

Variable speed characteristics of the KSB pump (KSB ETA pumps, n.d.).

Applying equations (7) and (8) to the specified pump yields the following empirical relations

h = 3.18 E - 6 N^{2} - 105600 q^{2}

(12)

\begin{array}{l} η = (- 0.00918 + 4.06 E - 6 N) + (166.194 + 0.02627 N) q \\ + (47153.43 - 25.1 N) q^{2} + (- 1.1 e 7 + 3781.79 N) q^{3} \end{array}

(13)

The pump data and equations (12) and (13) constitute the information needed to size the turbine and analyse the turbine/pump system. Table 3 shows the sizing results.

Table 3.

Turbine sizing results for Case II.

$ρ_{a}$ (kg/m³)	v (m/s)	H (m)	Turbine diameter d (mechanical transmission) (m)	Turbine diameter d (electrical transmission) (m)	C_P	$λ$
1.226	5.0	24	10.3	12.4	0.442	7.6

It can be shown that at a reference wind speed of 5.0 m/s, and a pump head of 24 m, the design Froude number, v/√(gh), is 0.33 and the corresponding gearbox transmission ratio is 51 for the electrical transmission case.

The mechanical and electrical transmission efficiencies were assumed fixed at 95%. The rated generator and motor efficiencies were estimated as 84%, for both components.

Case III: the Addison pump-based system

The third pump characteristics to analyse is the one reported in Addison (1964). The pump characteristics are shown in Figure 4.

Figure 4.

Variable speed characteristics of the ‘Addison’ pump (Addison H (1964)).

The wind turbine is the same as the one used in Cases I and II and shown in Figure 1.

By applying equations (7) and (8) to the case, the following head and efficiency expressions were obtained

h = 0.000037 N^{2} - 144 q^{2}

(14)

\begin{matrix} η_{p} = (- 0.005 + 0.00000658 N) + (19.52 - 0.0101 N) q \\ + (- 110.01 + 0.0779 N) q^{2} + (112.8 - 0.0855 N) q^{3} \end{matrix}

(15)

The basic pump data for this case are shown in Table 4 and the system sizing results are tabulated.

Table 4.

Turbine sizing results for the Addison pump.

$ρ_{a}$ (kg/m³)	V (m)	H (m)	d (mechanical) (m)	d (electrical) (m)	C_P	$λ$
1.226	7.00	33	34.4	38.1	0.442	7.6

At the adopted reference wind speed of 7.0 m/s, and a pump head of 33 m, the corresponding Froude number is 0.39 and the optimum transmission ratio value was 32 for the mechanical transmission and 36 for the electrical transmission.

Dependence of the optimum gearbox transmission ratio on Froude number

This section presents the analytical results obtained when each of the three centrifugal pumps is connected mechanically to the turbine. Constraining the turbine to perform at its peak power coefficient requires a variable transmission ratio, r_n. The system performance was analysed accordingly. The main results are shown in their dimensionless forms in Figures 5 to 7.

Figure 5.

Variation of the optimum transmission ratio with the wind pump’s Froude number, v/√(gh), of the Gambica pump.

Figure 6.

Variation of the optimum transmission ratio with the wind pump’s Froude number, v/√(gh), of the KSB pump.

Figure 7.

Variation of the optimum transmission ratio with the wind pump’s Froude number, v/√(gh), of the Addison pump.

It is clear that the optimum gearbox transmission ratio depends strongly on the wind pump’s Froude number, v/√(gh). For each pump, the analyses were performed at different pump heads. The dimensionless variations for the various head levels fall within a single curve and, therefore, independent of head. This important result has a positive effect on modelling and control. The control system shall deal with a single function linking the transmission ratio to the instantaneous Froude number value. Figure 6, for example, indicates that the control function for this case is

r_{n} = 17 F r^{- 0.8}

(16)

It is also important to note that the power coefficient C_P is kept constant at its peak value C_PM all through the continuous variable transmission (CVT) operation. The variation in the C_P.η value, as shown in Figure 8, is solely due to the variation in the efficiency η.

Figure 8.

Variation of the optimum C_PM.η value with the wind pump’s Froude number of the Gambica pump.

Regarding control of the optimized operation, this should in principle involve a control box that receives signals of measured wind velocity and water head, compute the corresponding Froude Number, use the system’s speed ratio-Froude number relationship, such as equation (16), to compute the optimum speed ratio then signal a step-motor to set the CVT to its optimized level. Such a control arrangement exists in multiples of industrial applications, particularly the auto industry (NISSAN XTRONIC CVT, 2018; Pesgens et al., 2003). Justification for cost would depend on the scale of power produced.

Performance of wind-powered electrical centrifugal pumps

The power train for the case where an intermediary electric power transmission is needed includes an induction generator following the gearbox and a cable transmitting the electrical power to an induction motor coupled to the centrifugal pump. The motor and generator part-load efficiencies were estimated in this work on basis of the IE2 High Efficiency ratings (IEC 60034-30-1) (Siemens Corporation, 2016). The net power reduction due to the three electrical stages, namely, the generator, transmission cable and motor ranges between 20% and 30%. The turbine diameter was corrected accordingly to compensate for the predicted extra losses. Figure 9 shows the resulting dependence of the net wind-to-water conversion factor on the wind pump’s Froude Number v/√(gh) for the turbine/Gambica-pump system.

Figure 9.

Variation of the net power conversion factor with Froude number for the electrical transmission option (Gambica pump).

From the control-system perspective, the most important variable is the optimum gearbox transmission ratio r_n. This was shown above as effectively dependent on a single dimensionless parameter, namely Fr. Once the turbine is sized for a given centrifugal pump, then the system can be made to perform optimally using a single control function of the form r_n = f(Fr). Figures 10 and 11 show the optimum CVT functions for the wind-driven Gambica and KSB pumps. The figures also compare the options optimized for the electrical and mechanical centrifugal pumps.

Figure 10.

Dependence of the transmission ratio on Froude number (the Gambica pump).

Figure 11.

Dependence of the transmission ratio on Froude number (the KSB pump).

Rated-speed operation of wind-powered electrical centrifugal pumps

The above analysis was subsequently extended to predict the system behaviour when the wind velocity is high enough as to tend to drive the pump speed above its rated maximum.

Considering the Gambica pump case, at around Fr = 0.32, the pump’s rotational speed is 1480 r/min and that, according to the manufacturer data, is presumably the maximum allowable for the pump. For Fr > 0.32, the pump rotational speed is to be constrained to 1480 r/min (Figures 12 and 13). This high Fr range is analogous to the rated constant generation power of wind generators.

Figure 12.

CVT performance extended into the rated speed range (Gambica pump).

Figure 13.

The variation of water capacity with the wind pump’s Froude Number extended into the rated speed range (Gambica pump).

The case for constant transmission ratio operation

The three wind pumping systems analysed in section ‘Demonstrations based on actual turbine and pump specifications’ demonstrated how to make use of the simple power-Froude number relationship to optimally match centrifugal pumps and wind turbines. To realize the full benefit of this result, the gearbox needs to be of the controlled-variable-transmission type, CVT.

Such a system can be too expensive for wind pumps of the traditional size. However, if large diameter powerful wind turbines are used for large-scale water supply or a pumped storage scheme, then such expenditure is probably worthwhile. It is also noted that the efficiency–speed relationship for centrifugal pumps is close, in principle, to that of electrical generators. This makes the efficient high-speed low-solidity turbine needed for a wind generator is a proper choice for wind-powered centrifugal pumps.

However, if the simpler system of a constant speed ratio is sought, then the above modelling is still of use and helps to make the proper selection of pump and turbine that gives the least sacrifice; in addition to an accurate choice of the constant transmission ratio. To demonstrate these points, the above analysis was extended to cover the performance of the studied wind pumps at selected fixed transmission ratios and compare the resulting performance to that of the CVT option.

Figure 14 illustrates the predicted performance of the Gambica pump coupled to the 40.2 m wind turbine described in section ‘Demonstrations based on actual turbine and pump specifications’. The results for the CVT operation reported in section ‘Demonstrations based on actual turbine and pump specifications’ are compared here to two constant speed ratio operations, namely, those with the r_n values 60 and 70.

Figure 14.

Effect of the transmission ratio r_n on net conversion efficiency for the Gambica pump.

The observations on these results:

At high Fr levels, that is, within the rated speed range, the performance for all considered transmission ratios converges to a single line.

The high transmission ratio is favourable within the low Fr range; lower wind speeds and/or higher lifts.

The low transmission ratio is favourable within the medium Fr range; medium wind speeds and/or medium lifts.

These observations fine-tune the common understanding that if a centrifugal pump is sized such that it matches a turbine at a single point then it would perform satisfactorily at all other wind speeds (Bergey, 1998). However, the discrepancy for relatively large wind pumps can be considerably large. The decision to fit a CVT or choose a particular fixed transmission ratio should then be based on the scheme’s economics.

Conclusion

Mathematical analysis of the performance of centrifugal pumps powered by wind turbines indicated that the wind pump’s Froude number v/√(gh) is the unique dimensionless parameter that determines the performance details of the system. Existence of such formulations helps in optimizing and simplifying control of such machines.

Use of continuous variable transmission (CVT) in wind pumps effectively improves the yield of wind pumps.

Depending on cost/benefit estimates, a constant transmission ratio gearbox can be preferred in some cases.

Wind-based water supply schemes can be effectively improved using large-scale wind turbines coupled directly or electrically to centrifugal pumps.

Large-scale wind turbines can be sized to power motor-driven centrifugal pumps that lift water to pumped storage facilities and simplify existing wind-dependent pumped storage systems.

Footnotes

Appendix 1 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) received no financial support for the research, authorship and/or publication of this article.

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