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A methodology to obtain the performance curves of windmills associated with piston pumps: application in water desalination systems by reverse osmosis


The use of windmills for the pumping of well water has appeared economically feasible in regions with average winds above 5 m/s, and when the water is used for human consumption and irrigation in arid rural zones, the importance of extracting the water from these wells efficiently and sustainably becomes more evident. However, concerning the state of Ceará/Brazil, only 18% of the drilled wells are freshwater wells, thus requiring additional energy for the utilization of desalination systems such as reverse osmosis. For that, the correct sizing of the pumping system associated with water desalination systems is essential, bearing in mind that the knowledge of the performance curves of the windmills, usually associated with piston pumps, is the “key factor” of the project. A brief analysis of the literature and websites of windmill suppliers shows that the performance curves of windmills associated with piston pumps are scarce, and the procedure to obtain them is not clear. This work’s novelty is to use the windmill’s power coefficient curve in association with the piston pumps to obtain the windpump performance curves, as pressure head versus volumetric flow rate in different wind conditions and sizes of windmills. The obtained curves were compared with other two curves found in the literature: one in a supplier’s catalog and another from the “rule of thumb”. The results showed a considerable divergence when calculating the root mean square deviation: 43.73% in relation to the supplier data average, and 125.76% in relation to the rule of thumb average results; however, the model proposed herein presented a limited operational range, which was not verified in the other two curves, which reproduced unlimited machines in terms of provided pressure head and flow rate. The unreal and asymptotic characteristics are more evident in the curves obtained by the rule of thumb, where the differences are also greater (125.76%) in relation to the proposed model. In terms of freshwater production, the model proposed herein reached, with an average wind speed of 6.5 m/s, and with a windmill of 6.1 m in diameter, a flow rate of up to 9.1 m\(^{3}\)/day, after going through the reverse osmosis system. The same condition with the rule of thumb’s model reached a flow rate of up to 11.3 m\(^{3}\)/day. Finally, the model proposed herein for obtaining windmill’s performance curves proved to be easy to implement and with realistic results, becoming an important design tool.

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\(A_{0}\) :

Transversal area of the windmill (\(\mathrm{m}^{2}\))

\(A_{b}\) :

Transversal area of the pump piston (\(\mathrm{m}^{2}\))

C :

Concentration of salts (\(\mathrm{mg}/\mathrm{L}\))

\(C_{a}\) :

Feed water salts concentration (\(\mathrm{mg}/\mathrm{L}\))

\(c_{p}\) :

Windmill power coefficient (−)

\(C_{pe}\) :

Permeate water salts concentration (\(\mathrm{mg}/\mathrm{L}\))

\(C_{r}\) :

Waste water salts concentration (\(\mathrm{mg}/\mathrm{L}\))

D :

Pipe diameter (\(\mathrm{m}\))

\(D_{j}\) :

j’s component’s diameter of Sect. 1 (\(\mathrm{m}\))

\(D_{k}\) :

k’s component’s diameter of Sect. 2 (\(\mathrm{m}\))

\(f_{j}\) :

Darcy friction coefficient of j’s components (−)

\(f_{p}\) :

Polarization factor of the membranes (−)

g :

Gravity acceleration (\(\mathrm{m}/\mathrm{s}^{2}\))

h :

Head (\(\mathrm{m}\))

\(h_{b}\) :

Pump’s head (\(\mathrm{m}\))

\(h_{i}\) :

Punctual i’s heads (\(\mathrm{m}\))

\(h_{j,pd}\) :

Major j’s losses of Sect. 1 (\(\mathrm{m}\))

\(h_{j,pl}\) :

Minor j’s losses of Sect. 1 (\(\mathrm{m}\))

\(h_{k,pd}\) :

Major k’s losses of Sect. 2 (\(\mathrm{m}\))

\(h_{k,pl}\) :

Minor k’s losses of Sect. 2 (\(\mathrm{m}\))

\(h_{sys}\) :

Total head required by the system (\(\mathrm{m}\))

\(h_{sys1}\) :

Head required by Sect. 1 (\(\mathrm{m}\))

\(h_{sys2}\) :

Head required by Sect. 2 (\(\mathrm{m}\))

\(K_{j}\) :

j’s component’s minor losses coefficients of Sect. 1 (−)

\(K_{k}\) :

k’s component’s minor losses coefficients of Sect. 2 (−)

\(K_{per}\) :

Permeability coefficient of the membranes (\(\mathrm{m}/(\mathrm{Pa.s})\))

\(L_{j}\) :

j’s pipes length of Sect. 1 (\(\mathrm{m}\))

\(L_{j,equ}\) :

j’s component’s minor losses equivalent length of Sect. 1 (\(\mathrm{m}\))

\(L_{k}\) :

k’s pipes length of Sect. 2 (\(\mathrm{m}\))

\(L_{k,equ}\) :

k’s component’s minor losses equivalent length of Sect. 1 (\(\mathrm{m}\))

n :

Rotation of the windmill’s rod actuation axis (\(-\))

P :

Manometric Static Pressure (\(\mathrm{Pa}\))

\(P_{i}\) :

Manometric Static Pressure of i’s points (\(\mathrm{Pa}\))

\(\overline{P}_{hb}\) :

Hydraulic power of the pump (W )

\(\overline{P}_{md}\) :

Mechanical power provided by the windmill (W )

\(\overline{P}_{mr}\) :

Mechanical power required by the pump (W)

Q :

Volumetric flowrate (\(\mathrm{m}^{3}/\mathrm{s}\))

\(Q_{a}\) :

Feed water flowrate (\(\mathrm{m}^{3}/\mathrm{day}\))

\(Q_{k}\) :

Volumetric flowrate into k’s components (\(\mathrm{m}^{3}/\mathrm{s}\))

\(Q_{p}\) :

Permeate flowrate (\(\mathrm{m}^{3}/\mathrm{s}\;or\;\mathrm{m}^{3}/\mathrm{day }\))

\(Q_{r}\) :

Waste water flowrate (\(\mathrm{m}^{3}/\mathrm{day}\))

\(Q_{9}\) :

Permeate flowrate (\(\mathrm{m}^{3}/\mathrm{s}\))

\(Q_{11}\) :

Waste flowrate (\(\mathrm{m}^{3}/\mathrm{s}\))

\(R_{0}\) :

Windmill radius (m)

s :

Course of the piston pump (m)

\(\overline{V}\) :

Average wind speed (\(\mathrm{m}/\mathrm{s}\) )

\(v_{i}\) :

Mean velocity into the i’s pipes (\(\mathrm{m}/\mathrm{s}\))

\(v_{j}\) :

Mean velocity into the j’s components of Sect. 1 (\(\mathrm{m}/\mathrm{s}\))

\(v_{k}\) :

Mean velocity into the k’s components of Sect. 2 (\(\mathrm{m}/\mathrm{s}\))

\(\beta\) :

Quality factor (−)

\(\Delta P\) :

Trans-membrane pressure difference (Pa)

\(\Delta P_{0}\) :

Trans-membrane pressure difference without the influence of the polarization factor (Pa)

\(\Delta S\) :

Area of the membrane (\(\mathrm{m}^{2}\))

\(\eta _{s}\) :

Volumetric yield of the piston pump (\(-\))

\(\gamma\) :

Specific weight of water (\(\mathrm{N}/\mathrm{m}^{3}\))

\(\lambda\) :

Tip speed rate of the blade (\(-\))

\(\omega\) :

Rotation of the windmill (\(\mathrm{rad}/\mathrm{s}\))

\(\pi\) :

Osmotic pressure (Pa)

\(\pi _{med}\) :

Average osmotic pressure (Pa)

\(\pi _{p}\) :

Osmotic pressure of the permeate (Pa)


Membrane 1


Membrane 2


Membrane 3


Membrane 4


Membrane 5


Polyvinyl chloride


T junction of pipes


Total Dissolved Solids


Tip speed rate of the blade (\(-\))


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We would like to thank the National Agency of Electric Energy - ANEEL for the encouragement in R&D - Research and Development, the Company Energia Pecém for the technological partnerships and financial support in the development of this work and the IFCE Campus Maracanaú for providing its laboratory facilities.

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Correspondence to Francisco Frederico dos Santos Matos.

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dos Santos Matos, F.F., Palombella, F.O., Dantas, P.H.S.M. et al. A methodology to obtain the performance curves of windmills associated with piston pumps: application in water desalination systems by reverse osmosis. J Braz. Soc. Mech. Sci. Eng. 43, 322 (2021).

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  • Windmills
  • Performance curves
  • Piston pumps
  • Head curves of windmill pumps