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Low technical wind pumping of high efficiency
Renewable Energy 24 (2001) 275–301 www.elsevier.nl/locate/renene
Technical note
Low technical wind pumping of high efficiency L.-C. Valde`s a
a,*
, K. Raniriharinosy
b
Groupe de Recherche Energies et Environnement, Universite´ de Valenciennes, BP311, 59304 Valenciennes Cedex, France b Universite´ de Fianarantsoa, BP 1294, 301 Fianarantsoa, Madagascar Received 6 October 2000; accepted 11 October 2000
Abstract Improving the conditions of water supply is an economic development factor of great consideration in poor countries. The wind power devices described in this article have been designed to give satisfaction in specific material conditions, notably those in Madagascar. By using low technical methods and in respecting the environment, these devices serve as the catalyst for immediate and long-term economic development. Three different versions of the device, corresponding to domestic, agricultural and light equipment needs, respectively, are worked out using a rationnal method of dimensioning. 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Improving the living standards of rural populations in developing countries rests largely on better control of the water supply. This would make daily life easier and improve hygiene and public health conditions. On an agricultural level, it would often increase land productivity and would enable the cultivated areas to be extended. Aid from developed countries has long taken the form of applying technically sophisticated solutions, judged to be the best. But we know of many cases where installations have been abandoned because there has been too great a difference between the techniques being used and those of the rural populations concerned. Today, more and more Non-Governmental Organisations involved in development aid have opted for a strategy of adapting the devices to technico–economic contexts, * Corresponding author. Tel.: +33-3-2751-1170; fax: +33-3-2751-1200. E-mail addresses: [email protected] (L.-C. Valde`s), (K. Raniriharinosy). 0960-1481/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 1 4 8 1 ( 0 0 ) 0 0 2 0 1 - 9
[email protected]
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Nomenclature Specifications V H V˙u
wind speed pumping height of lift usable volumetric flow
Maximal working point (point of maximum speed) Cp,M ⌳M
power coefficient speed ceofficient
Nominal working point (point of maximal efficiency) Cp,0 ⌳0 w
power coefficient speed coefficient speed of rotation
Geometry of the wind turbine rotor (Fig. 1) R L
radius height of the vane
Geometry of the pump piston (Fig. 5) a b B K c e d lf j
piston thickness piston width piston length B/b ratio piston stroke roller thickness design allowance leak length play between the piston and the fixed components
Roller eccentric (Fig. 6) da de E d db F Ne Np
shaft diameter eccentric diameter entre-axe between the two bearings design allowance glass marble diameter charge transmitted by the shaft number of marbles in the roller eccentric number of marbles in each bearing
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Limits of mechanical resistance slim flim gM Llim
maximum maximum maximum maximum
constraint exerted inside the shaft effort tolerated by a marble centrifugal acceleration in the vane height of the wind turbine
Transmission between the wind turbine and the pump hm
mechanical output
Physical properties r ra lf
volumic mass of water volumic mass of air global linear charge loss coefficient through leakage
with a view to obtaining results concerning development that are immediate as well as long-term. The authors of this article have attempted to apply these principles in contributing to a better control of the water supply in Madagascar. They have designed a wind pumping device of low technicality. The choice of the technical solutions was guided by two objectives. The first was to ensure the future installation of the pumping devices in as large an area as possible; the second was to end up with a device giving a reasonable energetic performance, according to the criteria in developed countries. This wind pumping device was then realised in collaboration with technicians from a Catholic Mission in Fianarantsoa (Madagascar), the CAPR (the Local Centre for Rural Development), in August 1999 (photograph in Fig. 1). By putting the prototype into action, the most suitable solutions concerning local know-how and resources were achieved. It highlighted some initial faults in the design that were then corrected. The final technical solutions are presented in Section 2. The performances of such devices depend on a great number of parameters. Only a systematic method of dimensioning should be able, right from the dimensioning stage, to ensure good energetic performance of the wind pumping answering to given specifications. A method is outlined in Sections 3 and 4 of the present article. Finally, several examples of such devices, answering the needs of rural populations in Madagascar, are given in Section 6. They result from applying the preceding method of determination and the numerical implementations explained in Section 5.
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Fig. 1.
Upper parts of the wind pumping device constructed in CAPR, August 1999.
2. Technical solutions 2.1. General Following the recommendations of the Harare declaration [1], the device destined to better control water supply would have to use one of the renewable energy sources: heat or photo-voltaic solar energy, biomass, wind, hydraulic or tide energy. This gives a wide choice of technical possibilities. The final choice is due to the great availability of wind energy as well as the way it can be used simply. But, to produce the desired beneficial effects, the wind pumping devices would have to satisfy two further conditions. First of all, they would have to be very cheap as the population that need them most also have the lowest buying power, and country people must be able to maintain and repair the devices. The second criterion was that the devices could be set up autonomously. The devices would have been constructed on the basis of know-how, tools and techniques available in a rural environment. This supposes equipment and accesories that are easy to obtain in country areas, as well as technical solutions that can be employed using rudimentary tools. The technical solutions described in this section have, for the most part, been used and tested on the prototype model constructed in collaboration with the CAPR technicians. 2.2. The device The overall architecture is shown in Fig. 2. The pump, which pulls the water up, is the central element in the system. To avoid priming problems, the pump is submerged in the water. The wind turbine, which captures the kinetic energy contained
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Fig. 2.
279
Overall architecture of wind pumping devices used for wells.
in the wind, is responsible for driving the pump. The tower supports the wind turbine in the face of the wind and provides the mechanical link between the wind turbine and the pump. The tower supports the mobile whole and guides its rotation round the vertical axis. It is built from bamboo stems and is no more than 3 m high, so it may be installed without specific elevation devices. An iron rod for reinforced concrete marks the vertical axis of the mobile whole. This iron rod drives all the elements in rotation: the wind turbine vanes and the mobile components of the pump. The mobile whole is hung from a wooden platform at the top of the tower. Its rotation is made smooth by a thrust bearing, with glass marbles and a ball race carved in a cylinder of hard wood. The iron rod ensures the power transmission through wooden cylinders ensuring the iron rod is guided inside a PVC tube. Special apparatus, eliminating unnecessary translations, enables transmission of the driving torque from the iron rod to the pump power take-off shaft, by minimising friction and doing away with any risk of the device jamming.
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2.3. The wind turbine More than a dozen different models of wind turbines are listed [2]. The criterion for choice was guided by the need for easy implementation and efficiency. It was necessary to select a wind turbine with a vertical axis as this eliminated directional problems of facing into the wind and the angle of return of the power take-off shaft. Choosing a machine with a slow speed of rotation did away with runaway problems due to strong winds as well. Finally, the wind turbine selected as that presenting the best compromise between its manufacture and easy maintenance and acceptable energetic performances was of the Savonius type, where the vanes of the rotor are two half-cylinders made to move around a vertical axis of rotation. The particularity of the half-cylinders recommended in this article comes from the choice of materials for manufacture: the principal part of the rotor is a bamboo frame. The vanes are made out of plaits of woven reeds. Using flexible bamboo imposes the cylindrical shape of the vanes (Fig. 1). 2.4. The pumps There are numerous different types of pumps. The choice of pump here is in part dictated by the choice of materials, wood in this case. The reason for this choice is the ease with which the wood planks and boards can be worked using traditional skills. The pumps are of a type able to attain the heights of lift corresponding to the depths of the largest wells: 20 to 30 m (the wells in Madagascar are dug by hand), with the speeds of rotation of the wind turbine: 1 to 3 turns per second in normal winds (the pumps must be driven directly by the wind turbine, to keep the technical solution simple and to minimise energy losses within the mechanism). Thus, the pumps are of the volumetric type. The pump here has a double-acting piston (Fig. 3). Its movement is in a plane perpendicular to the rotational axis of the wind turbine. The transformation of the rotational movement of the axis to a movement of alternative translation of the piston is produced by an eccentic shaft. Nonreturn valves control the flow. Using wood for the parts of the eccentric leads to the mechanism jamming. To solve this problem, the eccentric is made up of a roller on glass marbles, in line with the solution adopted for the thrust bearing (Section 2.2). Likewise, the shaft is equipped with two glass marble bearings. The impermeability of the nonreturn valves conditions the level of the height of lift. The valves with a glass marble and whose seat is carved out of a plastic tube supply a satisfactory technical solution. Outside the capital city in Madagascar, it is easier to obtain glass marbles as calibrated balls than steel ball bearings of a given diameter, so the former were chosen for this reason. The fact that they do not corrode also tipped the balance in favour of glass marbles.
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Fig. 3.
281
Exploded view of the pump.
2.5. Retained variants The leaks in the pump is the main limiting factor of the height of lift. Two pump variants, following the principle described in Fig. 3, still allow the main application to be covered. The first variant is a diaphragm pump. Impermeability is here made absolute by the use of a diaphragm cut into the air chamber. In this way, it is possible to attain great heights of lift, but due to the diaphragm’s limited capacity to change shape, the strokes are relatively low and these pumps have low rates of flow. The second variant is an open pump, designed to generate high rates of flow. To obtain large strokes, a diaphragm is no longer used, and to compensate for the inevitable leaks between the piston and the fixed parts of the pump, the piston section is made to be as large as possible. This type of pump is not capable of great heights of lift.
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3. Basic relations of dimensioning The present section is an inventory of the relations linking the different variables given at the beginning of the article, to each other. 3.1. Pump characteristics 3.1.1. Arranging equations The Fig. 4 presents the problem. It is supposed that the cross-section areas are dimensioned so the kinetic energy variations are negligible. Each element in the delivering circuit (index i) or leakage (index f) is modelled by a duct of hydraulic diameter di (or df), length li (or lf), cross-section si (or sf) and linear charge loss coefficient li (or lf). Singular charge loss on linking up the latter is noted zi (or zf). By applying Bernoulli’s theorem to the circuit carrying V˙ u (between the inside of the piston and the upper free surface), it gives
冋冘
ppist⫺patm⫽
i
li 1 l⫹ di s2i i
冘 册
Fig. 4.
i
r 1 z · V˙ 2⫹rg(H⫹h). s2i i 2 u
Data of the pumping modelling.
(1)
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In the open pump variant, there are leaks between the piston and the fixed parts of the pump. When applied to the part of the circuit carrying the leak flow rate V˙ f (between the inside of the piston and the lower water level), Bernoulli’s theorem gives
冋冘
ppist⫺patm⫽
f
冘 册
lf 1 l⫹ df s2f f
f
r 2 1 ˙ 2zf · Vf ⫹rgh. sf 2
(2)
The equation for mass balance links the above two rates of flow to the geometric flow rate V˙ generated by the piston movement by the relation: (3) V˙ ⫽V˙ u⫹V˙ f. In the case of diaphragm pumps, the relation at (3) becomes V˙ ⫽V˙ u. 3.1.2. Characteristic curve relative to the height of lift Eqs. (1) and (2) having the same lefthand side, they become gH⫽⫺aV˙ 2u⫹bV˙ 2f,
(3a)
(4)
with the subsidiary quantities a and b defined by: a⫽
1 2
and b⫽
1 2
冋冘 i
冋冘 f
冘 册 1 z s2i i
(5)
冘 册
(6)
li 1 l⫹ di s2i i
lf 1 l⫹ df s2f f
i
f
1 z . s2f f
As water delivering occurs during the backward and forward strokes of the doubleacting piston, the flow rate V˙ with the shaft driven at w is, on average over a period abc.w . V˙ ⫽ p
(7)
By transferring (7) in the V˙ u/V˙ trinomial obtained by eliminating V˙ f from the equation at (4) using the relation at (3), we have
冊 冢 冣 冢 冣 2
2
a V˙ u V˙ u gH ⫺1 ⫺ ⫽ . 2 abc.w b abc.w abc.w b p p p
冉
(8)
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3.1.3. Characteristic curve under pressure The piston is subjected to pressure ppist on the side facing the cavity of variable volume and on the other side, to pressure plow present at the lower water level. The force F is therefore F⫽ab(ppiston⫺plow).
(9)
The pressure present in the cavity is given, in particular, by Eq. (2). By eliminating the leak flow rate by using (3), and by using the relations at (7) and (9), we arrive at
冊 冢 冣
2
V˙ u F ⫺1 . ⫽ 2 abc.w abc.w abrb p p
冉
(10)
3.1.4. Pumping power ˙ p is given by the product of the force acting on the piston The pumping power W (relation (9)) by the the speed v of the piston. This last being driven by an eccentric revolving at the rotational speed ω, the piston speed can be written as wc v⫽ cos wt 2
(11)
The average pumping power is given by w 2p
冕
2p/w
|F.v| dt.
0
Using the relations at (9) and (11), we have ˙ p⫽F.wc. W 2p
(12)
3.2. Wind turbine 3.2.1. Radius The radius R is determined by a condition of mechanical resistance. This is expressed by means of the centrifugal acceleration g=w2R and is written as gⱕglim.
(13)
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Expressed by means of ⌳, defined by ⌳⫽
wR , V
(14)
the condition at (13) gives, with the maximum tolerated speed conditions, the searched for condition: (⌳MVM)2 . RⱕRlim⫽ glim
(15)
3.2.2. Rotational speed at the point of nominal functioning By applying the formula at (14) with the variables at the point of nominal functioning, we have w⫽
⌳0V0 . R
(16)
3.2.3. Rotor vane height ˙ p originates The height L is determined by expressing that the pumping power W ˙ w. According to the mechanical from the wind turbine, that supplies the power W losses due to the transmission, of output hm, between the wind turbine and the pump, we have, ˙ p⫽hmW ˙ w. W
(17)
˙ w is given traditionally, according to Cp, as: The power W ˙ w⫽Cp.raRLV3, W
(18)
˙ p is obtained from the formula at (12). By transferring this to the formula at and W (17), we obtain ⌳0.F.c L⫽ . 2p.hmCp,0.ra.R2V2
(19)
3.2.4. Maximum rotor vane height When w is fixed, the mechanical constraints produced by the centrifugal forces acting on the vanes of a wind turbine with vertical axis are a function of the slenderness defined by the ratio between the height L and the radius R [3]. Here, R being fixed at the maximum value Rlim defined by (15), the maximum height Llim can be found from the slenderness ratio for which the mechanical constraint reaches its maximum value.
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Thus, the condition of the vane resistance to the centrifugal forces: LⱕLlim can be written ⌳0.F.c ⱕL L⫽ 2p.hmCp,0.ra.R2V2 lim
(20)
3.3. Pump piston 3.3.1. Charge The charge F exerted by the pressure forces on the piston is determined by comparing the given characteristic curves regarding height of lift (formula (8)) and pressure (formula (10)). In remarking that a (charge losses from the flow reversal circuit) is very small compared to b (leak charge losses), we have F⫽abrgH.
(21)
3.3.2. Stroke The quantity 2d+2e+2db often occurs below. To simplify the writing of the formulas, it is taken that ⌬⫽2(d⫹e⫹db).
(22)
The piston stroke is determined by taking into account the spatial requirement of the roller eccentric system. The drawing in Fig. 5 allows us to write
Fig. 5.
Layout of the roller eccentric system in the stroke.
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c⫽b⫺2d⫺2e⫺2db⫺de.
287
(23)
According to the drawing in Fig. 6, we also have de⫽da⫹c.
(24)
By eliminating de from the formula (23) by means of (24), we obtain b−⌬−da . c⫽ 2
(25)
The stroke is a positive or nil quantity: cⱖ0. Expressed in terms of the relation (25), this condition provides the dimensioning constraint (26)
bⱖ⌬⫹da
Fig. 6. Longitudinal view of the eccentric shaft.
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3.3.3. Leak section The leaks occur in an annular cylinder, of rectangular shape. The hydraulic diameter df of the annular cylinder is given by df⫽2j.
(27)
Being a double-acting piston, the leaks occur over two preceding annular sections. Thus, we have sf⫽4j(a⫹b⫹2j).
(28)
The diagram in Fig. 5 allows us to establish the expression of length lf over which the leaks occur. By eliminating de by means of (24) then c by means of (25), we get lf⫽
(2K−1)b−⌬−da+4d . 4
(29)
The coefficient b defined at (6) can now be explained using the formulas (27), (28) and (29). By putting aside zf, we obtain lf (2K−1)b−⌬−da+4d . b⫽ 256j 3 (a+b+2j)2
(30)
The length of the leak is essentially positive (lfⱖ0); the relation (29) gives the dimensioning constraint (2K⫺1)bⱖ⌬⫹da⫺4d.
(31)
3.3.4. Useable volumetric flow Expression of V˙ u is obtained from the characteristic curve relative to pressure (formula (10)). By expressing the coefficient in the leak section (formula (30)), we have
V˙ u⫽
冢
abc.w 1⫹ p
冪
冣
256p2j 3(a+b+2j)2.F . rlf.ab(abc.w)2.((2K−1)b−⌬+4d−da)
(32)
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3.4. Roller eccentric 3.4.1. Mechanical constraint The diameter de is, here, taken to be identical to diameter da. This hypothesis lowers the real constraint. Arranging the equations depends on the diagram in Fig. 6. The theoretical modelling of the shaft load is that of a bending beam supported singly by two bearings. The beam has a circular cross-section of diameter da. The two bearings are spaced out by E. The charge F is exerted in the median section. Following this modelling, the constraint s produced by F is expressed as s⫽
FE . 4p 3 da 32
(33)
With the maximum mechanical constraint slim, the condition of resistance when bent: sⱕslim, is written s⫽
8 FE ⱕslim. p d 3a
(34)
3.4.2. Charge transmitted by the marbles It is supposed that F is transmitted to the shaft eccentric by the third of the total number of marbles Ne situated closest to the line of action of F. It is further supposed that each of the charged marbles is subjected to the same force f. Thus, Ne verifies
冉 冋册 冊
Ne F ⫽ int ⫹1 , 3 f
(35)
where int designates the integer value. The total number of marbles can also be expressed by a condition of spatial requirement, inferred from Fig. 5. By approximating the chord by the arc, we obtain the relation p(da+db) . Ne⫽ db
(36)
The expression of f: F f⫽ p da+db −1 3 db
(37)
is deduced from (35) and (36). The condition of mechanical resistance of the glass marbles: fⱕflim is then
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F f⫽ ⱕflim. p da+db −1 3 db
(38)
3.4.3. Entre-axe Fig. 6 shows the two cases concerning the eccentric shaft. It illustrates the existing relation E⫽
再
2db+d,
if a⬍db
(39)
a+db+d, if a⬎db
3.4.4. Number of marbles in the bearings Determining Ne and Np is rigorously worked out at this point. An elementary trigonometric calculation leads to
Np⫽int
冤
p
冉
db arcsin da+db
冊冥
.
(40)
3.4.5. Number of marbles in the eccentric Adapted to the case of the eccentric by replacing da by de, formula (40) gives, using (24)
冤
Ne⫽int
冉
p
db arcsin da+db+c
冊冥
.
(41)
4. Rational method of dimensioning The method established here allows us to systematically determine the parameters of wind pumping devices answering given specifications and the relations in Section 3.
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4.1. General data of the problem 4.1.1. Unknowns The problem is precisely to find all the dimensions enabling a wind pumping device to draw up V˙ u, over H, using wind with speed V. The variables governing a wind pumping device taken as unknowns are: F, da, a, b, c. 4.1.2. Basic relations The relations considered in priority convey the specifications imposed on the wind pumping device: relations at (21) (height of lift H) and (32) (useable volumetric flow V˙ u). The relation (25) between b and c is also considered. The other relations are expressed by inequalities. The first ones are in relation to the viability of the device and express the conditions of mechanical resistance of the pump shaft (relation (34)), the marbles (relation (38)) and the frame of the wind turbine rotor (relation (20)). These inequalities will be achieved in the least restrictive way as possible. Three more inequalities express the building constraints: the positivity of the pump shaft diameter (daⱖ0), the piston stroke (relation (26)) and the length of the leak section (relation (31)). 4.1.3. Arranging the mathemetical problem The problem is a priori to find F, da, a, b, c verifying the system of equations (21), (32) and (25), inequations (34), (38) and (20) and inequations daⱖ0, (26) and (31). This is simplified by eliminating c in the relations at (21), (32) and (20) by using the relation (25). Moreover, to make the mathematical analysis of the problem easier, a graphical representation is used. Plane (da, F) is adopted. Finally, the problem acknowledges F, da, a and b as unknowns. In (da, F) plane, the relations (21) and (32), that must be now written as curve equations F=F(da), become: F⫽ab.rgH and F⫽
冉
(42)
冊
rlfw2 a3b ((2K⫺1)b⫺⌬⫹4d⫺da) 2 3 1024p j (a+b+2j)2
⫺bda
冊
冉
b(b⫺⌬)⫺
2p V˙ u w a
(43)
2
whereas inequations (34), (38) and (20) express as:
冉 冊 冉 冊 冉 冊
0ⱕsⱕslim⇔0ⱕFⱕ 0ⱕfⱕflim⇔0ⱕFⱕ
p slim 3 .d a , 8 E
p flim p .da⫹ ⫺1 .flim, 3 db 3
(44) (45)
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0ⱕLⱕLlim⇔0ⱕFⱕ
4p.hmCp,0·ra·R2V2.Llim . ((b−⌬)−da).⌳0
(46)
The fact that the constraints (44) to (46) must be achieved in the least restrictive way possible (Section 4.1.2), necessitates that the searched for point be located on the area of definition boundary, that is to say, on that of the equation curves s=slim, L=Llim, or f=flim with the lowest ordinate. This is expressed by
F⫽ min {0ⱕda}
冉 冊 冉 冊 冉 冊
冦
冧
p slim 3 .d a , 8 E
p flim p .d + −1 .flim, . 3 db a 3
(47)
4p.hmCp,0.ra.R2V2.Llim ((b−⌬)−da).⌳0
Finally, inequalities daⱖ0, (26) and (31) will be written daⱖ0, cⱖ0 and lfⱖ0 ⇔ 0ⱕdaⱕD⫽min{(b⫺⌬),(2K⫺1)b⫺⌬⫹4d}.
(48)
The relation (47) brings an additional constraint to the equations (42) and (43), that the four unknowns F, da, a, b must verify. The degree of freedom that remains is retained and, consequently, the unknowns will be determined according to one amongst them. 4.1.4. Status of the different variables The vacant degree of freedom (Section 4.1.3) is attributed to the variable a with the result that the method aims to supply the relations b=b(a), F=F(a) and da=da(a). The choice of the definitive quadruplet (F, da, a, b) calls for the intervention of the criteria presented in Section 5.2. Variable b acts as principal unknown. Working in the (da, F) plane means considering F and da as generic unknowns, linked to the solution point. 4.2. Resolution — open pump 4.2.1. The formula giving b on the curve f=flim By expressing that the relations (42) and (43) must be verified on the boundary curve (45), we obtain, after eliminating da and F:
冉 冊 冊冉
ab.rgH⫽
冊冉冉 冉
rlfw2 a3b 1024p2j 3 (a+b+2j)2
⫺1 db b(b⫺⌬)⫺
2K⫺1⫺
冊 冉 冊 冊冊
冉
3 adb.rgH 3 b⫺⌬⫹4d⫺ p flim p
2p V˙ u 3 adb.rgH 3 b⫹ ⫺1 db ⫺b w a p flim p
2
.
(49)
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By using the b value solution of (49), the equation of the boundary curve at (45) provides the searched for value of da: da⫽
冉 冊
3 adb.rgH 3 b⫹ ⫺1 db. p flim p
(50)
4.2.2. The formula giving b on the curve L=Llim By expressing that the relations (42) and (43) must be verified on the boundary curve (46), we obtain, after eliminating da and F: ab.rgH⫽
冉
冊冉
rlfw2 a3b 4p.hmCp,0.ra.R2V2.Llim 2p V˙ u ⫺ 2 3 2 1024p j (a+b+2j) a.rgH.⌳0 w a
⫺1)b⫹4d⫹
4p.hmCp,0.ra.R2V2.Llim 1 a.rgH.⌳0 b
冊
冊冉 2
2(K
(51)
By using the b value solution of (51), the equation of the boundary curve (46) provides the searched for value of da: 4p.hmCp,0.raR2V2.Llim 1 da⫽b⫺⌬⫺ . a.rgH.⌳0 b
(52)
4.2.3. The formula giving b on the curve s=slim By expressing that the relations (42) and (43) must be verified on the boundary curve (44), we obtain, after eliminating da and F:
冉
ab.rgH⫽
rlfw2 a 3b 2 3 1024p j (a+b+2j)2
冢 冢
冣
冪p
8 aE.rgH 1/3 .b slim
(2K⫺1)b⫺⌬⫹4d⫺ 3
2p V˙ u 3 b(b⫺⌬)⫺ ⫺ w a
冊
冣
2
冪p
8 aE.rgH 4/3 .b . slim
(53)
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By using the b value solution of (53), the equation of the boundary curve at (44) provides the searched for value of da: da⫽ 3
冪p
8 aE.rgH 1/3 .b . slim
(54)
4.2.4. Searching for b The formulas (49), (51) and (53) give the values of b for which the point of coordinates (da, F) verifies the specifications and is situated on one of the curves f=flim, L=Llim or s=slim, respectively. To know if these values of b are solutions, it is necessary to verify if the relation (47) is verified for the b values and corresponding da and F values considered. A great number of cases a priori leading to impossible solutions is eliminated in searching for the values of b verifying the building constraint bⱖ0.
(55)
Finally, the building constraints cⱖ0 and lfⱖ0 must be satisfied. It is therefore necessary to make da verify the inequality (48). 4.3. Resolution — diaphragm pump The formula (42) is not modified: F⫽ab.rgH
(42a)
The other formulas presented in Sections 4.1 and 4.2 become simplified (no leaks between the piston and the fixed parts). That replacing formula (43) is obtained by writing into (7) that V˙ f is nil (formula (3a)). After eliminating c using (25), we obtain 2p V˙ u da⫽b⫺⌬⫺ . w ab
(43a)
The formulas (42a) and (43a) and the unchanged relations (44) to (46) allow us to establish that the value of the parameter b, which gives a solution (1) on the curve f=flim, is given by ab.rgH⫽
冉
冊冉 冊
2p V˙ u p p flim b⫺⌬⫺ ⫹ ⫺1 flim. 3 db w ab 3
(49a)
(2) on the curve L=Llim is given by the relation independent of a and b V˙ u hmCp,0raR2V2Llim , rgH. ⫽ w ⌳0
(51a)
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(3) on the curve s=slim, is given by ab.rgH⫽
冉
冊
2p V˙ u 3 p slim b⫺⌬⫺ . 8 E w ab
(53a)
The method described in Section 4.2.4 is applied using the new formulas (49a), (51a) and (53a). 5. Numerical implementation The numerical examples described in Section 6 were obtained using computer programmes written on a pocket computer TI-89. The algorithm of the resolution method is described in Section 5.1. All the other information is given in the appendix of [4]. Section 5.2 provides the criteria allowing the choice of the definitive solution. Section 5.3 gives the numerical values of the constants used in the numerical examples in Section 6. 5.1. Algorithm 5.1.1. Stage 0 The values are set for: H, V˙ u, V, ra, r, g, ⌳0, Cp,0, hm, lf, slim, flim, db, j, d, e, K. 5.1.2. Stage 1 Calculation of R by taking the value Rlim (formula (15)), of w (formula (16)), and Llim (Section 3.2.4). 5.1.3. Stage 2 Given a value of a, then calculation of E using the formula (39). 5.1.4. Stage 3 Calculation of b(a) and da (a): for b verifying (55), — calculation of b by (49) or by (49a), of da by (50) then of Fs by (44), of Ff by (45) and of FL by (46). The solution, if it exists, is retained if the inequality (48) is verified and if Ff=min{Fs, Ff, FL}, according to relation (47). — calculation of b by (51) or by (51a), of da by (52) then of Fs by (44), of Ff by (45) and of FL by (46). The solution, if it exists, is retained if the inequality (48) is verified and if FL=min{Fs, Ff,FL}, according to relation (47). — calculation of b by (53) or by (53a), of da by (54) then of Fs by (44), of Ff by (45) and of FL by (46). The solution, if it exists, is retained if the inequality (48) is verified and if Fs=min{Fs, Ff, FL}, according to relation (47). 5.1.5. Stage 4 Calculation of c(a) by (25) and of F by (42) for each candidate value of b. Returning to the stage 2 as many times as is necessary.
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5.1.6. Stage 5 The final choice of the value of a is made (Section 5.2). The quantities that remain to be returned are: L (formula (19)), de (formula (24)), Ne (formula (41), Np (formula (40)), E (formula (39)). 5.2. Criteria for choosing the definitive quadruplet 5.2.1. Manufacture Raw materials can be economised if the dimensions of the various elements of the pump are close to (in fact, slighly less than) the standard dimensions of the boards and planks. Regarding Madagascar, the standard dimensions of the boards are 3 cm×15 cm×500 cm. Planks of 6 cm×15 cm×400 cm are common, but there are also those with dimensions 15 cm×15 cm×400 cm. 5.2.2. Piston stroke The diaphragm is the sensitive element in a diaphragm pump and it is to be expected that it will tear periodically due to fatigue. To limit the frequency of replacement, the piston stroke is set at a low value. A maximum value of 5 mm has been chosen here. 5.2.3. Torque transmission The maximal torque transmitted by the wind turbine is given by Mt⫽
F.c . 2
(56)
while the torsional torque that the iron rod transmits is given, according to the maximum elastic constraint, by the formula Mt⫽
p.telasf3 , 32
(57)
With a maximum elastic constraint telas of 144.106 Pa, the correspondence for maximal torques and standard diameters of iron rod for reinforced concrete is given in Table 1
Table 1 Maximal torques and standard diameters of iron rod φ (mm) Mt (m.N)
6 3.05
8 7.2
10 14.1
12 24.4
14 38.8
16 57.9
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5.3. Numerical values common to the applications 5.3.1. Physical constants The physical constants used in the numerical applications are: r=1,000 kg/m3 and g=9.81 m/s2. The volumic mass ra=1.2 kg/m3 corresponds to air at a pressure of 1 atm and a temperature of 20°C. 5.3.2. The Savonius wind turbine point of functioning Its power coefficient and its speed coefficient define the nominal point of functioning of the Savonius wind turbine. The values chosen here are: Cp,0=0.25 and ⌳0=0.8 (see [2]). 5.3.3. Design data The glass marble diameter is, with a dimensional tolerance in the order of a tenth of a mm, db=0.016 m. Wood machining techniques would make it difficult to create the functioning play between the piston and the fixed parts less than j=0.001 m. Finally, the rules of good design led us to choose the values d=0.01 m and e=0.01 m. 5.3.4. Functioning data The measurements carried out on the pump prototype enabled us to determine that lf=0.106. The marbles thrust bearing, the marble bearings and the torque transmission apparatus described in Section 2.2 should allow us to have the value hm=0.65. 5.3.5. Mechanical resistance The limit of the charge imposed on the pump shaft comes from the maximum mechanical constraint. A reasonable value for this is slim=25.106 Pa. The low value flim=100 N allowed for the glass marbles is meant to retard rupture through fatigue. The mechanical resistance of the bamboo frame of the wind turbine must enable it to work at the lowest gale force (7 Beaufort). The calculation is carried out for the functioning conditions: VM=18 m/s, ⌳M=1.7, gM=750 m/s2 and gives R=1.25 m. The slenderness ratio L/R of the wind turbine rotor conditions its mechanical resistance to centrifugal forces [3]. An approximate relation of 2 has been retained. 6. Some suitable devices for Madagascar 6.1. Specifications 6.1.1. Ordinary wells A common depth for a household well in Madagascar is of the order of 7.5 m. If we assume that the water will be stocked in a slightly raised reservoir, the typical
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height of flow reversal is 12 m. Ordinary wells are dug not far from the homes. A reasonable value regarding wind speed is 4 m/s, corresponding to Beaufort scale 3. The production of such wells is of the order of several hundred liters per day. It is possible to anticipate a useable volumetric flow of 1,000 l/day for the pump. The wind turbine of the device for ordinary wells is then usually subject to irregular wind conditions. In order to draw up the amount required for an entire day over short periods, the useable volumetric flow must be oversized. An overflow system must be planned to let the surplus of water drawn up flow back to the well. Doing this supplies an elementary process for oxygenating the water from the well and purifying it microbially. 6.1.2. Irrigation of the rice farming plains The situation in question here is that of rice farming valleys in Madagascar. These are, for the most part, open to winds and are found in the high plains region. Rice is cultivated in terraces, and dykes not greater than a man’s height separate the fields. A system of channels leads the water from a river at a point upstream. In this way, a significant area remains more often than not untouched by irrigation and is then made unsuitable for agriculture. Irrigating against gravity, from the lower terraces up to the higher ones, could be implemented using the wind pumping devices outlined in this section. The water must be drawn up over the height of the dykes separating the terraces, say 1.5 m. Besides this, we have often a stiff breeze, corresponding to degree 5 on the Beaufort scale, say V=9 m/s. Finally, the water requirement of V˙ u=1 l/s is set by the continuing fictitious peak flow. In order to limit maintenance (of the diaphragm) and to generate the above large useable volumetric flow, the pump is of an open pump type. 6.1.3. Basic hydraulic equipment The height of water elevation is the main drawback for this type of application because it is set at a value around the depth of deep wells, 24 m. This height means it is necessary to use a diaphragm pump. The flow rate is fixed at V˙ u=2000 l/d. Other applications may be envisaged with this type of pump: drawing water up from a river or a lake to the top of a hill, circulating water from a far-off source through a pipe, or irrigating mountain sides. As is the case for irrigation (Section 6.1.2), wind pumping may be joined in series or in parallel, to increase the height of water elevation or the rate of flow, respectively. As the device will probably be set up in open ground, the wind conditions are presumed to be fairly good: V=7 m/s (Beaufort scale 4). 6.2. Dimensioning Applying the dimensioning algorithm (Section 5.1) using the above specifications leads to results according to a. They are given in Tables 2–4. The values in the tables above show there exist sets of values close to the standard
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Table 2 Devices for ordinary wells a (m) b (m) c (m) d (m) F (N)
0.016 0.121 0.0073 0.034 227
0.018 0.124 0.0063 0.039 263
0.020 0.129 0.0055 0.046 302
0.022 0.134 0.0048 0.052 347
0.024 0.140 0.0042 0.060 396
0.026 0.148 0.0037 0.068 452
0.028 0.157 0.0032 0.078 516
Table 3 Irrigation devices a (m) b (m) c (m) d (m) F (N)
0.05 0.358 0.123 0.039 263
0.075 0.316 0.096 0.053 349
0.100 0.298 0.080 0.067 439
0.125 0.292 0.069 0.081 536
0.150 0.292 0.061 0.098 645
0.175 0.297 0.055 0.116 766
0.017 0.207 0.0046 0.126 829
0.018 0.225 0.0040 0.145 953
0.019 0.247 0.0035 0.168 1105
0.020 0.275 0.0029 0.197 1295
Table 4 Devices for basic hydraulic equipment a (m) b (m) c (m) d (m) F (N)
0.013 0.163 0.008 0.075 497
0.015 0.181 0.006 0.097 635
Table 5 Dimensions of planks and boards a (m) Ordinary wells Irrigation devices Hydraulic equipment
0.022 0.100 0.018
b (m) 0.134 0.298 0.225
c (m) 0.005 0.080 0.004
da (m) 0.052 0.067 0.145
dimensions of the planks and boards (Section 5.2.1). The chosen sets are shown in Table 5. The rest of the determination leads to Table 6. According to formula (56), the recommended diameters of the iron rod are as shown in Table 7.
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Table 6 Remaining parameters
Ordinary wells Irrigation devices Hydraulic equipment
K
R (m)
L (m)
E (m)
1.5
1.25 1.25 1.25
2 2 2
0.048 0.126 0.044
dc(m) 0.057 0.145 0.149
Nc
Np
14 31 32
13 16 31
Table 7 Recommended diameters of iron rod Iron rod diameter (mm) Ordinary wells Irrigation devices Hydraulic equipment
6 8 12
7. Conclusions The wind pumping devices described here have been designed using solutions of low technicality. An advantage of these devices is their low cost (cost price of the whole prototype estimated at about 45 $US) and the simplicity of production and maintenance. Our aim is their large-scale installation in the countryside of developing countries having technico–economic characteristics similar to those of Madagascar. The wind pumping devices suggested can give good energetic performances if they are adapted in the best way to the water supply requirements they are meant to satisfy. A rational dimensioning method is presented. The application of this method gives three types of devices answering the needs of a great number of areas of daily life in Madagascar: the water supply from ordinary wells, drawing up water or its circulation from an accessible pumping point (river or lake) to the point of use, helping valleys open to the winds through irrigation against gravity.
Acknowledgements The authors would like to express their gratitude to the Brothers of the Catholic Mission, in charge of the CAPR, Fianarantsoa, for the trust they put into their initiative and the means they put at their disposal. They would also like to pay tribute to the enthusiasm and the competence of the CAPR technicians.
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References [1] Organisation des Nations Unies, UNESCO, Sommet solaire mondial, De´ claration de Harare sur l’e´ nergie solaire et le de´ veloppement durable, 17 septembre 1996. [2] Martin J. Energies e´ oliennes. Techniques de l’Inge´ nieur, traite´ Ge´ nie e´ nerge´ tique B8585:1–21, 1995. [3] Menet J-L, Valde`s L-C, Me´ nart B. A comparative calculation of the wind turbine capacities on the basis of the L-s criterion. Renewable Energy 2001;22:491–506. [4] Valde`s L-C. Dispositifs e´ oliens de pompage d’eau adapte´ s a` Madagascar. Partie 4, Exemples. RR GREEn 2000-4, March 2000.
Technical note
Low technical wind pumping of high efficiency L.-C. Valde`s a
a,*
, K. Raniriharinosy
b
Groupe de Recherche Energies et Environnement, Universite´ de Valenciennes, BP311, 59304 Valenciennes Cedex, France b Universite´ de Fianarantsoa, BP 1294, 301 Fianarantsoa, Madagascar Received 6 October 2000; accepted 11 October 2000
Abstract Improving the conditions of water supply is an economic development factor of great consideration in poor countries. The wind power devices described in this article have been designed to give satisfaction in specific material conditions, notably those in Madagascar. By using low technical methods and in respecting the environment, these devices serve as the catalyst for immediate and long-term economic development. Three different versions of the device, corresponding to domestic, agricultural and light equipment needs, respectively, are worked out using a rationnal method of dimensioning. 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Improving the living standards of rural populations in developing countries rests largely on better control of the water supply. This would make daily life easier and improve hygiene and public health conditions. On an agricultural level, it would often increase land productivity and would enable the cultivated areas to be extended. Aid from developed countries has long taken the form of applying technically sophisticated solutions, judged to be the best. But we know of many cases where installations have been abandoned because there has been too great a difference between the techniques being used and those of the rural populations concerned. Today, more and more Non-Governmental Organisations involved in development aid have opted for a strategy of adapting the devices to technico–economic contexts, * Corresponding author. Tel.: +33-3-2751-1170; fax: +33-3-2751-1200. E-mail addresses: [email protected] (L.-C. Valde`s), (K. Raniriharinosy). 0960-1481/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 1 4 8 1 ( 0 0 ) 0 0 2 0 1 - 9
[email protected]
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Nomenclature Specifications V H V˙u
wind speed pumping height of lift usable volumetric flow
Maximal working point (point of maximum speed) Cp,M ⌳M
power coefficient speed ceofficient
Nominal working point (point of maximal efficiency) Cp,0 ⌳0 w
power coefficient speed coefficient speed of rotation
Geometry of the wind turbine rotor (Fig. 1) R L
radius height of the vane
Geometry of the pump piston (Fig. 5) a b B K c e d lf j
piston thickness piston width piston length B/b ratio piston stroke roller thickness design allowance leak length play between the piston and the fixed components
Roller eccentric (Fig. 6) da de E d db F Ne Np
shaft diameter eccentric diameter entre-axe between the two bearings design allowance glass marble diameter charge transmitted by the shaft number of marbles in the roller eccentric number of marbles in each bearing
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Limits of mechanical resistance slim flim gM Llim
maximum maximum maximum maximum
constraint exerted inside the shaft effort tolerated by a marble centrifugal acceleration in the vane height of the wind turbine
Transmission between the wind turbine and the pump hm
mechanical output
Physical properties r ra lf
volumic mass of water volumic mass of air global linear charge loss coefficient through leakage
with a view to obtaining results concerning development that are immediate as well as long-term. The authors of this article have attempted to apply these principles in contributing to a better control of the water supply in Madagascar. They have designed a wind pumping device of low technicality. The choice of the technical solutions was guided by two objectives. The first was to ensure the future installation of the pumping devices in as large an area as possible; the second was to end up with a device giving a reasonable energetic performance, according to the criteria in developed countries. This wind pumping device was then realised in collaboration with technicians from a Catholic Mission in Fianarantsoa (Madagascar), the CAPR (the Local Centre for Rural Development), in August 1999 (photograph in Fig. 1). By putting the prototype into action, the most suitable solutions concerning local know-how and resources were achieved. It highlighted some initial faults in the design that were then corrected. The final technical solutions are presented in Section 2. The performances of such devices depend on a great number of parameters. Only a systematic method of dimensioning should be able, right from the dimensioning stage, to ensure good energetic performance of the wind pumping answering to given specifications. A method is outlined in Sections 3 and 4 of the present article. Finally, several examples of such devices, answering the needs of rural populations in Madagascar, are given in Section 6. They result from applying the preceding method of determination and the numerical implementations explained in Section 5.
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Fig. 1.
Upper parts of the wind pumping device constructed in CAPR, August 1999.
2. Technical solutions 2.1. General Following the recommendations of the Harare declaration [1], the device destined to better control water supply would have to use one of the renewable energy sources: heat or photo-voltaic solar energy, biomass, wind, hydraulic or tide energy. This gives a wide choice of technical possibilities. The final choice is due to the great availability of wind energy as well as the way it can be used simply. But, to produce the desired beneficial effects, the wind pumping devices would have to satisfy two further conditions. First of all, they would have to be very cheap as the population that need them most also have the lowest buying power, and country people must be able to maintain and repair the devices. The second criterion was that the devices could be set up autonomously. The devices would have been constructed on the basis of know-how, tools and techniques available in a rural environment. This supposes equipment and accesories that are easy to obtain in country areas, as well as technical solutions that can be employed using rudimentary tools. The technical solutions described in this section have, for the most part, been used and tested on the prototype model constructed in collaboration with the CAPR technicians. 2.2. The device The overall architecture is shown in Fig. 2. The pump, which pulls the water up, is the central element in the system. To avoid priming problems, the pump is submerged in the water. The wind turbine, which captures the kinetic energy contained
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Fig. 2.
279
Overall architecture of wind pumping devices used for wells.
in the wind, is responsible for driving the pump. The tower supports the wind turbine in the face of the wind and provides the mechanical link between the wind turbine and the pump. The tower supports the mobile whole and guides its rotation round the vertical axis. It is built from bamboo stems and is no more than 3 m high, so it may be installed without specific elevation devices. An iron rod for reinforced concrete marks the vertical axis of the mobile whole. This iron rod drives all the elements in rotation: the wind turbine vanes and the mobile components of the pump. The mobile whole is hung from a wooden platform at the top of the tower. Its rotation is made smooth by a thrust bearing, with glass marbles and a ball race carved in a cylinder of hard wood. The iron rod ensures the power transmission through wooden cylinders ensuring the iron rod is guided inside a PVC tube. Special apparatus, eliminating unnecessary translations, enables transmission of the driving torque from the iron rod to the pump power take-off shaft, by minimising friction and doing away with any risk of the device jamming.
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2.3. The wind turbine More than a dozen different models of wind turbines are listed [2]. The criterion for choice was guided by the need for easy implementation and efficiency. It was necessary to select a wind turbine with a vertical axis as this eliminated directional problems of facing into the wind and the angle of return of the power take-off shaft. Choosing a machine with a slow speed of rotation did away with runaway problems due to strong winds as well. Finally, the wind turbine selected as that presenting the best compromise between its manufacture and easy maintenance and acceptable energetic performances was of the Savonius type, where the vanes of the rotor are two half-cylinders made to move around a vertical axis of rotation. The particularity of the half-cylinders recommended in this article comes from the choice of materials for manufacture: the principal part of the rotor is a bamboo frame. The vanes are made out of plaits of woven reeds. Using flexible bamboo imposes the cylindrical shape of the vanes (Fig. 1). 2.4. The pumps There are numerous different types of pumps. The choice of pump here is in part dictated by the choice of materials, wood in this case. The reason for this choice is the ease with which the wood planks and boards can be worked using traditional skills. The pumps are of a type able to attain the heights of lift corresponding to the depths of the largest wells: 20 to 30 m (the wells in Madagascar are dug by hand), with the speeds of rotation of the wind turbine: 1 to 3 turns per second in normal winds (the pumps must be driven directly by the wind turbine, to keep the technical solution simple and to minimise energy losses within the mechanism). Thus, the pumps are of the volumetric type. The pump here has a double-acting piston (Fig. 3). Its movement is in a plane perpendicular to the rotational axis of the wind turbine. The transformation of the rotational movement of the axis to a movement of alternative translation of the piston is produced by an eccentic shaft. Nonreturn valves control the flow. Using wood for the parts of the eccentric leads to the mechanism jamming. To solve this problem, the eccentric is made up of a roller on glass marbles, in line with the solution adopted for the thrust bearing (Section 2.2). Likewise, the shaft is equipped with two glass marble bearings. The impermeability of the nonreturn valves conditions the level of the height of lift. The valves with a glass marble and whose seat is carved out of a plastic tube supply a satisfactory technical solution. Outside the capital city in Madagascar, it is easier to obtain glass marbles as calibrated balls than steel ball bearings of a given diameter, so the former were chosen for this reason. The fact that they do not corrode also tipped the balance in favour of glass marbles.
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Fig. 3.
281
Exploded view of the pump.
2.5. Retained variants The leaks in the pump is the main limiting factor of the height of lift. Two pump variants, following the principle described in Fig. 3, still allow the main application to be covered. The first variant is a diaphragm pump. Impermeability is here made absolute by the use of a diaphragm cut into the air chamber. In this way, it is possible to attain great heights of lift, but due to the diaphragm’s limited capacity to change shape, the strokes are relatively low and these pumps have low rates of flow. The second variant is an open pump, designed to generate high rates of flow. To obtain large strokes, a diaphragm is no longer used, and to compensate for the inevitable leaks between the piston and the fixed parts of the pump, the piston section is made to be as large as possible. This type of pump is not capable of great heights of lift.
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3. Basic relations of dimensioning The present section is an inventory of the relations linking the different variables given at the beginning of the article, to each other. 3.1. Pump characteristics 3.1.1. Arranging equations The Fig. 4 presents the problem. It is supposed that the cross-section areas are dimensioned so the kinetic energy variations are negligible. Each element in the delivering circuit (index i) or leakage (index f) is modelled by a duct of hydraulic diameter di (or df), length li (or lf), cross-section si (or sf) and linear charge loss coefficient li (or lf). Singular charge loss on linking up the latter is noted zi (or zf). By applying Bernoulli’s theorem to the circuit carrying V˙ u (between the inside of the piston and the upper free surface), it gives
冋冘
ppist⫺patm⫽
i
li 1 l⫹ di s2i i
冘 册
Fig. 4.
i
r 1 z · V˙ 2⫹rg(H⫹h). s2i i 2 u
Data of the pumping modelling.
(1)
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In the open pump variant, there are leaks between the piston and the fixed parts of the pump. When applied to the part of the circuit carrying the leak flow rate V˙ f (between the inside of the piston and the lower water level), Bernoulli’s theorem gives
冋冘
ppist⫺patm⫽
f
冘 册
lf 1 l⫹ df s2f f
f
r 2 1 ˙ 2zf · Vf ⫹rgh. sf 2
(2)
The equation for mass balance links the above two rates of flow to the geometric flow rate V˙ generated by the piston movement by the relation: (3) V˙ ⫽V˙ u⫹V˙ f. In the case of diaphragm pumps, the relation at (3) becomes V˙ ⫽V˙ u. 3.1.2. Characteristic curve relative to the height of lift Eqs. (1) and (2) having the same lefthand side, they become gH⫽⫺aV˙ 2u⫹bV˙ 2f,
(3a)
(4)
with the subsidiary quantities a and b defined by: a⫽
1 2
and b⫽
1 2
冋冘 i
冋冘 f
冘 册 1 z s2i i
(5)
冘 册
(6)
li 1 l⫹ di s2i i
lf 1 l⫹ df s2f f
i
f
1 z . s2f f
As water delivering occurs during the backward and forward strokes of the doubleacting piston, the flow rate V˙ with the shaft driven at w is, on average over a period abc.w . V˙ ⫽ p
(7)
By transferring (7) in the V˙ u/V˙ trinomial obtained by eliminating V˙ f from the equation at (4) using the relation at (3), we have
冊 冢 冣 冢 冣 2
2
a V˙ u V˙ u gH ⫺1 ⫺ ⫽ . 2 abc.w b abc.w abc.w b p p p
冉
(8)
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3.1.3. Characteristic curve under pressure The piston is subjected to pressure ppist on the side facing the cavity of variable volume and on the other side, to pressure plow present at the lower water level. The force F is therefore F⫽ab(ppiston⫺plow).
(9)
The pressure present in the cavity is given, in particular, by Eq. (2). By eliminating the leak flow rate by using (3), and by using the relations at (7) and (9), we arrive at
冊 冢 冣
2
V˙ u F ⫺1 . ⫽ 2 abc.w abc.w abrb p p
冉
(10)
3.1.4. Pumping power ˙ p is given by the product of the force acting on the piston The pumping power W (relation (9)) by the the speed v of the piston. This last being driven by an eccentric revolving at the rotational speed ω, the piston speed can be written as wc v⫽ cos wt 2
(11)
The average pumping power is given by w 2p
冕
2p/w
|F.v| dt.
0
Using the relations at (9) and (11), we have ˙ p⫽F.wc. W 2p
(12)
3.2. Wind turbine 3.2.1. Radius The radius R is determined by a condition of mechanical resistance. This is expressed by means of the centrifugal acceleration g=w2R and is written as gⱕglim.
(13)
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Expressed by means of ⌳, defined by ⌳⫽
wR , V
(14)
the condition at (13) gives, with the maximum tolerated speed conditions, the searched for condition: (⌳MVM)2 . RⱕRlim⫽ glim
(15)
3.2.2. Rotational speed at the point of nominal functioning By applying the formula at (14) with the variables at the point of nominal functioning, we have w⫽
⌳0V0 . R
(16)
3.2.3. Rotor vane height ˙ p originates The height L is determined by expressing that the pumping power W ˙ w. According to the mechanical from the wind turbine, that supplies the power W losses due to the transmission, of output hm, between the wind turbine and the pump, we have, ˙ p⫽hmW ˙ w. W
(17)
˙ w is given traditionally, according to Cp, as: The power W ˙ w⫽Cp.raRLV3, W
(18)
˙ p is obtained from the formula at (12). By transferring this to the formula at and W (17), we obtain ⌳0.F.c L⫽ . 2p.hmCp,0.ra.R2V2
(19)
3.2.4. Maximum rotor vane height When w is fixed, the mechanical constraints produced by the centrifugal forces acting on the vanes of a wind turbine with vertical axis are a function of the slenderness defined by the ratio between the height L and the radius R [3]. Here, R being fixed at the maximum value Rlim defined by (15), the maximum height Llim can be found from the slenderness ratio for which the mechanical constraint reaches its maximum value.
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Thus, the condition of the vane resistance to the centrifugal forces: LⱕLlim can be written ⌳0.F.c ⱕL L⫽ 2p.hmCp,0.ra.R2V2 lim
(20)
3.3. Pump piston 3.3.1. Charge The charge F exerted by the pressure forces on the piston is determined by comparing the given characteristic curves regarding height of lift (formula (8)) and pressure (formula (10)). In remarking that a (charge losses from the flow reversal circuit) is very small compared to b (leak charge losses), we have F⫽abrgH.
(21)
3.3.2. Stroke The quantity 2d+2e+2db often occurs below. To simplify the writing of the formulas, it is taken that ⌬⫽2(d⫹e⫹db).
(22)
The piston stroke is determined by taking into account the spatial requirement of the roller eccentric system. The drawing in Fig. 5 allows us to write
Fig. 5.
Layout of the roller eccentric system in the stroke.
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c⫽b⫺2d⫺2e⫺2db⫺de.
287
(23)
According to the drawing in Fig. 6, we also have de⫽da⫹c.
(24)
By eliminating de from the formula (23) by means of (24), we obtain b−⌬−da . c⫽ 2
(25)
The stroke is a positive or nil quantity: cⱖ0. Expressed in terms of the relation (25), this condition provides the dimensioning constraint (26)
bⱖ⌬⫹da
Fig. 6. Longitudinal view of the eccentric shaft.
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3.3.3. Leak section The leaks occur in an annular cylinder, of rectangular shape. The hydraulic diameter df of the annular cylinder is given by df⫽2j.
(27)
Being a double-acting piston, the leaks occur over two preceding annular sections. Thus, we have sf⫽4j(a⫹b⫹2j).
(28)
The diagram in Fig. 5 allows us to establish the expression of length lf over which the leaks occur. By eliminating de by means of (24) then c by means of (25), we get lf⫽
(2K−1)b−⌬−da+4d . 4
(29)
The coefficient b defined at (6) can now be explained using the formulas (27), (28) and (29). By putting aside zf, we obtain lf (2K−1)b−⌬−da+4d . b⫽ 256j 3 (a+b+2j)2
(30)
The length of the leak is essentially positive (lfⱖ0); the relation (29) gives the dimensioning constraint (2K⫺1)bⱖ⌬⫹da⫺4d.
(31)
3.3.4. Useable volumetric flow Expression of V˙ u is obtained from the characteristic curve relative to pressure (formula (10)). By expressing the coefficient in the leak section (formula (30)), we have
V˙ u⫽
冢
abc.w 1⫹ p
冪
冣
256p2j 3(a+b+2j)2.F . rlf.ab(abc.w)2.((2K−1)b−⌬+4d−da)
(32)
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3.4. Roller eccentric 3.4.1. Mechanical constraint The diameter de is, here, taken to be identical to diameter da. This hypothesis lowers the real constraint. Arranging the equations depends on the diagram in Fig. 6. The theoretical modelling of the shaft load is that of a bending beam supported singly by two bearings. The beam has a circular cross-section of diameter da. The two bearings are spaced out by E. The charge F is exerted in the median section. Following this modelling, the constraint s produced by F is expressed as s⫽
FE . 4p 3 da 32
(33)
With the maximum mechanical constraint slim, the condition of resistance when bent: sⱕslim, is written s⫽
8 FE ⱕslim. p d 3a
(34)
3.4.2. Charge transmitted by the marbles It is supposed that F is transmitted to the shaft eccentric by the third of the total number of marbles Ne situated closest to the line of action of F. It is further supposed that each of the charged marbles is subjected to the same force f. Thus, Ne verifies
冉 冋册 冊
Ne F ⫽ int ⫹1 , 3 f
(35)
where int designates the integer value. The total number of marbles can also be expressed by a condition of spatial requirement, inferred from Fig. 5. By approximating the chord by the arc, we obtain the relation p(da+db) . Ne⫽ db
(36)
The expression of f: F f⫽ p da+db −1 3 db
(37)
is deduced from (35) and (36). The condition of mechanical resistance of the glass marbles: fⱕflim is then
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F f⫽ ⱕflim. p da+db −1 3 db
(38)
3.4.3. Entre-axe Fig. 6 shows the two cases concerning the eccentric shaft. It illustrates the existing relation E⫽
再
2db+d,
if a⬍db
(39)
a+db+d, if a⬎db
3.4.4. Number of marbles in the bearings Determining Ne and Np is rigorously worked out at this point. An elementary trigonometric calculation leads to
Np⫽int
冤
p
冉
db arcsin da+db
冊冥
.
(40)
3.4.5. Number of marbles in the eccentric Adapted to the case of the eccentric by replacing da by de, formula (40) gives, using (24)
冤
Ne⫽int
冉
p
db arcsin da+db+c
冊冥
.
(41)
4. Rational method of dimensioning The method established here allows us to systematically determine the parameters of wind pumping devices answering given specifications and the relations in Section 3.
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4.1. General data of the problem 4.1.1. Unknowns The problem is precisely to find all the dimensions enabling a wind pumping device to draw up V˙ u, over H, using wind with speed V. The variables governing a wind pumping device taken as unknowns are: F, da, a, b, c. 4.1.2. Basic relations The relations considered in priority convey the specifications imposed on the wind pumping device: relations at (21) (height of lift H) and (32) (useable volumetric flow V˙ u). The relation (25) between b and c is also considered. The other relations are expressed by inequalities. The first ones are in relation to the viability of the device and express the conditions of mechanical resistance of the pump shaft (relation (34)), the marbles (relation (38)) and the frame of the wind turbine rotor (relation (20)). These inequalities will be achieved in the least restrictive way as possible. Three more inequalities express the building constraints: the positivity of the pump shaft diameter (daⱖ0), the piston stroke (relation (26)) and the length of the leak section (relation (31)). 4.1.3. Arranging the mathemetical problem The problem is a priori to find F, da, a, b, c verifying the system of equations (21), (32) and (25), inequations (34), (38) and (20) and inequations daⱖ0, (26) and (31). This is simplified by eliminating c in the relations at (21), (32) and (20) by using the relation (25). Moreover, to make the mathematical analysis of the problem easier, a graphical representation is used. Plane (da, F) is adopted. Finally, the problem acknowledges F, da, a and b as unknowns. In (da, F) plane, the relations (21) and (32), that must be now written as curve equations F=F(da), become: F⫽ab.rgH and F⫽
冉
(42)
冊
rlfw2 a3b ((2K⫺1)b⫺⌬⫹4d⫺da) 2 3 1024p j (a+b+2j)2
⫺bda
冊
冉
b(b⫺⌬)⫺
2p V˙ u w a
(43)
2
whereas inequations (34), (38) and (20) express as:
冉 冊 冉 冊 冉 冊
0ⱕsⱕslim⇔0ⱕFⱕ 0ⱕfⱕflim⇔0ⱕFⱕ
p slim 3 .d a , 8 E
p flim p .da⫹ ⫺1 .flim, 3 db 3
(44) (45)
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0ⱕLⱕLlim⇔0ⱕFⱕ
4p.hmCp,0·ra·R2V2.Llim . ((b−⌬)−da).⌳0
(46)
The fact that the constraints (44) to (46) must be achieved in the least restrictive way possible (Section 4.1.2), necessitates that the searched for point be located on the area of definition boundary, that is to say, on that of the equation curves s=slim, L=Llim, or f=flim with the lowest ordinate. This is expressed by
F⫽ min {0ⱕda}
冉 冊 冉 冊 冉 冊
冦
冧
p slim 3 .d a , 8 E
p flim p .d + −1 .flim, . 3 db a 3
(47)
4p.hmCp,0.ra.R2V2.Llim ((b−⌬)−da).⌳0
Finally, inequalities daⱖ0, (26) and (31) will be written daⱖ0, cⱖ0 and lfⱖ0 ⇔ 0ⱕdaⱕD⫽min{(b⫺⌬),(2K⫺1)b⫺⌬⫹4d}.
(48)
The relation (47) brings an additional constraint to the equations (42) and (43), that the four unknowns F, da, a, b must verify. The degree of freedom that remains is retained and, consequently, the unknowns will be determined according to one amongst them. 4.1.4. Status of the different variables The vacant degree of freedom (Section 4.1.3) is attributed to the variable a with the result that the method aims to supply the relations b=b(a), F=F(a) and da=da(a). The choice of the definitive quadruplet (F, da, a, b) calls for the intervention of the criteria presented in Section 5.2. Variable b acts as principal unknown. Working in the (da, F) plane means considering F and da as generic unknowns, linked to the solution point. 4.2. Resolution — open pump 4.2.1. The formula giving b on the curve f=flim By expressing that the relations (42) and (43) must be verified on the boundary curve (45), we obtain, after eliminating da and F:
冉 冊 冊冉
ab.rgH⫽
冊冉冉 冉
rlfw2 a3b 1024p2j 3 (a+b+2j)2
⫺1 db b(b⫺⌬)⫺
2K⫺1⫺
冊 冉 冊 冊冊
冉
3 adb.rgH 3 b⫺⌬⫹4d⫺ p flim p
2p V˙ u 3 adb.rgH 3 b⫹ ⫺1 db ⫺b w a p flim p
2
.
(49)
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By using the b value solution of (49), the equation of the boundary curve at (45) provides the searched for value of da: da⫽
冉 冊
3 adb.rgH 3 b⫹ ⫺1 db. p flim p
(50)
4.2.2. The formula giving b on the curve L=Llim By expressing that the relations (42) and (43) must be verified on the boundary curve (46), we obtain, after eliminating da and F: ab.rgH⫽
冉
冊冉
rlfw2 a3b 4p.hmCp,0.ra.R2V2.Llim 2p V˙ u ⫺ 2 3 2 1024p j (a+b+2j) a.rgH.⌳0 w a
⫺1)b⫹4d⫹
4p.hmCp,0.ra.R2V2.Llim 1 a.rgH.⌳0 b
冊
冊冉 2
2(K
(51)
By using the b value solution of (51), the equation of the boundary curve (46) provides the searched for value of da: 4p.hmCp,0.raR2V2.Llim 1 da⫽b⫺⌬⫺ . a.rgH.⌳0 b
(52)
4.2.3. The formula giving b on the curve s=slim By expressing that the relations (42) and (43) must be verified on the boundary curve (44), we obtain, after eliminating da and F:
冉
ab.rgH⫽
rlfw2 a 3b 2 3 1024p j (a+b+2j)2
冢 冢
冣
冪p
8 aE.rgH 1/3 .b slim
(2K⫺1)b⫺⌬⫹4d⫺ 3
2p V˙ u 3 b(b⫺⌬)⫺ ⫺ w a
冊
冣
2
冪p
8 aE.rgH 4/3 .b . slim
(53)
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By using the b value solution of (53), the equation of the boundary curve at (44) provides the searched for value of da: da⫽ 3
冪p
8 aE.rgH 1/3 .b . slim
(54)
4.2.4. Searching for b The formulas (49), (51) and (53) give the values of b for which the point of coordinates (da, F) verifies the specifications and is situated on one of the curves f=flim, L=Llim or s=slim, respectively. To know if these values of b are solutions, it is necessary to verify if the relation (47) is verified for the b values and corresponding da and F values considered. A great number of cases a priori leading to impossible solutions is eliminated in searching for the values of b verifying the building constraint bⱖ0.
(55)
Finally, the building constraints cⱖ0 and lfⱖ0 must be satisfied. It is therefore necessary to make da verify the inequality (48). 4.3. Resolution — diaphragm pump The formula (42) is not modified: F⫽ab.rgH
(42a)
The other formulas presented in Sections 4.1 and 4.2 become simplified (no leaks between the piston and the fixed parts). That replacing formula (43) is obtained by writing into (7) that V˙ f is nil (formula (3a)). After eliminating c using (25), we obtain 2p V˙ u da⫽b⫺⌬⫺ . w ab
(43a)
The formulas (42a) and (43a) and the unchanged relations (44) to (46) allow us to establish that the value of the parameter b, which gives a solution (1) on the curve f=flim, is given by ab.rgH⫽
冉
冊冉 冊
2p V˙ u p p flim b⫺⌬⫺ ⫹ ⫺1 flim. 3 db w ab 3
(49a)
(2) on the curve L=Llim is given by the relation independent of a and b V˙ u hmCp,0raR2V2Llim , rgH. ⫽ w ⌳0
(51a)
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(3) on the curve s=slim, is given by ab.rgH⫽
冉
冊
2p V˙ u 3 p slim b⫺⌬⫺ . 8 E w ab
(53a)
The method described in Section 4.2.4 is applied using the new formulas (49a), (51a) and (53a). 5. Numerical implementation The numerical examples described in Section 6 were obtained using computer programmes written on a pocket computer TI-89. The algorithm of the resolution method is described in Section 5.1. All the other information is given in the appendix of [4]. Section 5.2 provides the criteria allowing the choice of the definitive solution. Section 5.3 gives the numerical values of the constants used in the numerical examples in Section 6. 5.1. Algorithm 5.1.1. Stage 0 The values are set for: H, V˙ u, V, ra, r, g, ⌳0, Cp,0, hm, lf, slim, flim, db, j, d, e, K. 5.1.2. Stage 1 Calculation of R by taking the value Rlim (formula (15)), of w (formula (16)), and Llim (Section 3.2.4). 5.1.3. Stage 2 Given a value of a, then calculation of E using the formula (39). 5.1.4. Stage 3 Calculation of b(a) and da (a): for b verifying (55), — calculation of b by (49) or by (49a), of da by (50) then of Fs by (44), of Ff by (45) and of FL by (46). The solution, if it exists, is retained if the inequality (48) is verified and if Ff=min{Fs, Ff, FL}, according to relation (47). — calculation of b by (51) or by (51a), of da by (52) then of Fs by (44), of Ff by (45) and of FL by (46). The solution, if it exists, is retained if the inequality (48) is verified and if FL=min{Fs, Ff,FL}, according to relation (47). — calculation of b by (53) or by (53a), of da by (54) then of Fs by (44), of Ff by (45) and of FL by (46). The solution, if it exists, is retained if the inequality (48) is verified and if Fs=min{Fs, Ff, FL}, according to relation (47). 5.1.5. Stage 4 Calculation of c(a) by (25) and of F by (42) for each candidate value of b. Returning to the stage 2 as many times as is necessary.
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5.1.6. Stage 5 The final choice of the value of a is made (Section 5.2). The quantities that remain to be returned are: L (formula (19)), de (formula (24)), Ne (formula (41), Np (formula (40)), E (formula (39)). 5.2. Criteria for choosing the definitive quadruplet 5.2.1. Manufacture Raw materials can be economised if the dimensions of the various elements of the pump are close to (in fact, slighly less than) the standard dimensions of the boards and planks. Regarding Madagascar, the standard dimensions of the boards are 3 cm×15 cm×500 cm. Planks of 6 cm×15 cm×400 cm are common, but there are also those with dimensions 15 cm×15 cm×400 cm. 5.2.2. Piston stroke The diaphragm is the sensitive element in a diaphragm pump and it is to be expected that it will tear periodically due to fatigue. To limit the frequency of replacement, the piston stroke is set at a low value. A maximum value of 5 mm has been chosen here. 5.2.3. Torque transmission The maximal torque transmitted by the wind turbine is given by Mt⫽
F.c . 2
(56)
while the torsional torque that the iron rod transmits is given, according to the maximum elastic constraint, by the formula Mt⫽
p.telasf3 , 32
(57)
With a maximum elastic constraint telas of 144.106 Pa, the correspondence for maximal torques and standard diameters of iron rod for reinforced concrete is given in Table 1
Table 1 Maximal torques and standard diameters of iron rod φ (mm) Mt (m.N)
6 3.05
8 7.2
10 14.1
12 24.4
14 38.8
16 57.9
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5.3. Numerical values common to the applications 5.3.1. Physical constants The physical constants used in the numerical applications are: r=1,000 kg/m3 and g=9.81 m/s2. The volumic mass ra=1.2 kg/m3 corresponds to air at a pressure of 1 atm and a temperature of 20°C. 5.3.2. The Savonius wind turbine point of functioning Its power coefficient and its speed coefficient define the nominal point of functioning of the Savonius wind turbine. The values chosen here are: Cp,0=0.25 and ⌳0=0.8 (see [2]). 5.3.3. Design data The glass marble diameter is, with a dimensional tolerance in the order of a tenth of a mm, db=0.016 m. Wood machining techniques would make it difficult to create the functioning play between the piston and the fixed parts less than j=0.001 m. Finally, the rules of good design led us to choose the values d=0.01 m and e=0.01 m. 5.3.4. Functioning data The measurements carried out on the pump prototype enabled us to determine that lf=0.106. The marbles thrust bearing, the marble bearings and the torque transmission apparatus described in Section 2.2 should allow us to have the value hm=0.65. 5.3.5. Mechanical resistance The limit of the charge imposed on the pump shaft comes from the maximum mechanical constraint. A reasonable value for this is slim=25.106 Pa. The low value flim=100 N allowed for the glass marbles is meant to retard rupture through fatigue. The mechanical resistance of the bamboo frame of the wind turbine must enable it to work at the lowest gale force (7 Beaufort). The calculation is carried out for the functioning conditions: VM=18 m/s, ⌳M=1.7, gM=750 m/s2 and gives R=1.25 m. The slenderness ratio L/R of the wind turbine rotor conditions its mechanical resistance to centrifugal forces [3]. An approximate relation of 2 has been retained. 6. Some suitable devices for Madagascar 6.1. Specifications 6.1.1. Ordinary wells A common depth for a household well in Madagascar is of the order of 7.5 m. If we assume that the water will be stocked in a slightly raised reservoir, the typical
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height of flow reversal is 12 m. Ordinary wells are dug not far from the homes. A reasonable value regarding wind speed is 4 m/s, corresponding to Beaufort scale 3. The production of such wells is of the order of several hundred liters per day. It is possible to anticipate a useable volumetric flow of 1,000 l/day for the pump. The wind turbine of the device for ordinary wells is then usually subject to irregular wind conditions. In order to draw up the amount required for an entire day over short periods, the useable volumetric flow must be oversized. An overflow system must be planned to let the surplus of water drawn up flow back to the well. Doing this supplies an elementary process for oxygenating the water from the well and purifying it microbially. 6.1.2. Irrigation of the rice farming plains The situation in question here is that of rice farming valleys in Madagascar. These are, for the most part, open to winds and are found in the high plains region. Rice is cultivated in terraces, and dykes not greater than a man’s height separate the fields. A system of channels leads the water from a river at a point upstream. In this way, a significant area remains more often than not untouched by irrigation and is then made unsuitable for agriculture. Irrigating against gravity, from the lower terraces up to the higher ones, could be implemented using the wind pumping devices outlined in this section. The water must be drawn up over the height of the dykes separating the terraces, say 1.5 m. Besides this, we have often a stiff breeze, corresponding to degree 5 on the Beaufort scale, say V=9 m/s. Finally, the water requirement of V˙ u=1 l/s is set by the continuing fictitious peak flow. In order to limit maintenance (of the diaphragm) and to generate the above large useable volumetric flow, the pump is of an open pump type. 6.1.3. Basic hydraulic equipment The height of water elevation is the main drawback for this type of application because it is set at a value around the depth of deep wells, 24 m. This height means it is necessary to use a diaphragm pump. The flow rate is fixed at V˙ u=2000 l/d. Other applications may be envisaged with this type of pump: drawing water up from a river or a lake to the top of a hill, circulating water from a far-off source through a pipe, or irrigating mountain sides. As is the case for irrigation (Section 6.1.2), wind pumping may be joined in series or in parallel, to increase the height of water elevation or the rate of flow, respectively. As the device will probably be set up in open ground, the wind conditions are presumed to be fairly good: V=7 m/s (Beaufort scale 4). 6.2. Dimensioning Applying the dimensioning algorithm (Section 5.1) using the above specifications leads to results according to a. They are given in Tables 2–4. The values in the tables above show there exist sets of values close to the standard
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Table 2 Devices for ordinary wells a (m) b (m) c (m) d (m) F (N)
0.016 0.121 0.0073 0.034 227
0.018 0.124 0.0063 0.039 263
0.020 0.129 0.0055 0.046 302
0.022 0.134 0.0048 0.052 347
0.024 0.140 0.0042 0.060 396
0.026 0.148 0.0037 0.068 452
0.028 0.157 0.0032 0.078 516
Table 3 Irrigation devices a (m) b (m) c (m) d (m) F (N)
0.05 0.358 0.123 0.039 263
0.075 0.316 0.096 0.053 349
0.100 0.298 0.080 0.067 439
0.125 0.292 0.069 0.081 536
0.150 0.292 0.061 0.098 645
0.175 0.297 0.055 0.116 766
0.017 0.207 0.0046 0.126 829
0.018 0.225 0.0040 0.145 953
0.019 0.247 0.0035 0.168 1105
0.020 0.275 0.0029 0.197 1295
Table 4 Devices for basic hydraulic equipment a (m) b (m) c (m) d (m) F (N)
0.013 0.163 0.008 0.075 497
0.015 0.181 0.006 0.097 635
Table 5 Dimensions of planks and boards a (m) Ordinary wells Irrigation devices Hydraulic equipment
0.022 0.100 0.018
b (m) 0.134 0.298 0.225
c (m) 0.005 0.080 0.004
da (m) 0.052 0.067 0.145
dimensions of the planks and boards (Section 5.2.1). The chosen sets are shown in Table 5. The rest of the determination leads to Table 6. According to formula (56), the recommended diameters of the iron rod are as shown in Table 7.
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Table 6 Remaining parameters
Ordinary wells Irrigation devices Hydraulic equipment
K
R (m)
L (m)
E (m)
1.5
1.25 1.25 1.25
2 2 2
0.048 0.126 0.044
dc(m) 0.057 0.145 0.149
Nc
Np
14 31 32
13 16 31
Table 7 Recommended diameters of iron rod Iron rod diameter (mm) Ordinary wells Irrigation devices Hydraulic equipment
6 8 12
7. Conclusions The wind pumping devices described here have been designed using solutions of low technicality. An advantage of these devices is their low cost (cost price of the whole prototype estimated at about 45 $US) and the simplicity of production and maintenance. Our aim is their large-scale installation in the countryside of developing countries having technico–economic characteristics similar to those of Madagascar. The wind pumping devices suggested can give good energetic performances if they are adapted in the best way to the water supply requirements they are meant to satisfy. A rational dimensioning method is presented. The application of this method gives three types of devices answering the needs of a great number of areas of daily life in Madagascar: the water supply from ordinary wells, drawing up water or its circulation from an accessible pumping point (river or lake) to the point of use, helping valleys open to the winds through irrigation against gravity.
Acknowledgements The authors would like to express their gratitude to the Brothers of the Catholic Mission, in charge of the CAPR, Fianarantsoa, for the trust they put into their initiative and the means they put at their disposal. They would also like to pay tribute to the enthusiasm and the competence of the CAPR technicians.
L.-C. Valde`s, K. Raniriharinosy / Renewable Energy 24 (2001) 275–301
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