Output power and power losses estimation for an

overshot water wheel

Emanuele Quaranta∗, Roberto Revelli

Politecnico di Torino, DIATI (Department of Environment, Land and InfrastructureEngineering).

Corso Duca degli Abruzzi 24, 10129, Torino, Italia

Abstract

Thanks to a new sensibility to renewable energy and to local and smart

electricity production, traditional water wheels are regarded again as a clean

and accessible way for pico-micro hydro electricity generation, especially in

presence of very low heads and small flowrates. In particular, among the

different kinds of water wheels, the overshot ones exploit the lowest flowrates

with the highest efficiency (efficiency up to 85-90%).

Therefore, in order to determine the performance characteristics and to

estimate the power losses and the output power for overshot water wheels,

theoretical and experimental analyses are here developed. By a numerical

optimization process, the power losses and the mechanical output power are

theoretically quantified. A critical angular velocity is also identified (about

65% of the runaway speed). When the wheel rotation speed approaches the

critical velocity, volumetric losses at the top of the wheel begin to increase

linearly with the rotational velocity, while the other power losses start to

∗Corresponding authorEmail address: emanuele.quaranta@polito.it (Emanuele Quaranta)

Preprint submitted to Renewable Energy June 16, 2016

decrease, due to the lower amount of water available to the wheel.

Keywords:

Hydropower; mechanical power; micro hydro; overshot water wheel; power

losses.

1. Introduction1

For the next decades, the International Energy Agency estimates a sig-2

nificant increase in energy demand, developing some possible scenarios [1].3

A first scenario foresees an overall increase in energy demand, mainly in4

China and India. According to this model, fossil fuels will remain the main5

sources for meeting the global energy demand: coal will present the great-6

est growth in absolute terms, while oil may remain the most widely used7

fuel. Such a situation would lead to an increase in the level of emissions. A8

second possible scenario is, instead, the possibility of a long-term transition9

towards a model of clean and renewable energy and sustainable develop-10

ment: an important opportunity may be represented by energy production11

through water wheels.12

Water wheels were introduced more than 2000 years ago, to pump wa-13

ter and to grind cereals. The most ancient water wheel was built with14

a vertical axle for transmit power to the millstone; although the vertical15

axle technology was the simplest, its efficiency was quite low. The first16

kind with horizontal axle was the stream water wheel, already described17

by Vitruvius in 27 BC. In stream water wheels the stream passes near the18

bottom of the wheel, impinging against the blades, and the kinetic energy19

of the stream is mainly exploited. Instead, in the overshot, breastshot and20

2

undershot wheels the potential energy of water is the main source of power.21

The water flow enters into the cells at the top of the wheel or at about22

the same height of the rotation axle, or at the bottom, respectively; it is23

then carried inside the buckets, where it acts by its weight on the blades,24

up to the tailrace, where the buckets empty. Although the stream water25

wheel has very low efficiency (the most of the water kinetic energy is dis-26

sipated in the impact against the blades), it has been incorrectly preferred27

over the gravity water wheels for centuries. This mistake was due both to28

the theory on jets developed in 1704 by Antoine Parent, which limited the29

hydraulic efficiency of all water wheels to just 14.8% (but his analysis was30

mathematically incorrect and not applicable to all types of water wheels),31

and to the simplest and most intuitive technology of stream water wheels.32

In the middle of the Eighteenth century, John Smeaton published experi-33

mental data and demonstrated the better efficiency of the overshot water34

wheel over the efficiency of the stream one [2]. Indeed, the exploitation35

of potential energy instead of kinetic energy leads to higher efficiency, but36

higher heads are necessary [3] [4] [5] [6] [7] [8]. In particular, overshot water37

wheels are suitable in sites with heads of 2.5-10 m and small flowrates (unit38

width flowrates Q < 0.1−0.2 m2s−1) and they can reach constant efficiency39

of about 80-90% for a wide range of flowrates [3].40

By the time, theories and manufacturing methods of water wheels im-41

proved [9] [10] [11] [12] [13] [14] [15] [16] [17]; the most developed technology42

took a significant leap into the realm of turbines at the end of the Nineteenth43

century and water wheels development ceased.44

In these years, thanks to a new sensibility to renewable energy and45

3

to local and smart electricity production, traditional water wheels are re-46

garded again as a clean and accessible way for pico-micro hydro electricity47

generation (installed power lower than 5 KW and 100 KW respectively),48

especially in presence of very low heads. Water wheels are a sustainable49

and economic technology, since their construction is simpler over turbines,50

their environmental impact is lower, the payback periods are faster and51

practically there is no public resistance to their installation. They may also52

improve the local economy by promoting tourism and cultural activities,53

in addition to crop grinding and electrical production. Water wheels may54

not be a strategical solution for large scale renewable energy generation,55

but they may be a suitable method for decentralized electricity generation56

and for a smart land use. For this reason, in the last decade the related57

scientific research has seen a revival [3] [4] [5] [6] [7] [8]. In particular, in58

[18] a detailed work has been conducted on a breastshot water wheel, illus-59

trating a modern theoretical model (and comparing it with some past ones)60

and experimental results. Similarly, the present work aims to fill the gap61

of engineering information on overshot water wheels. Indeed, although a62

large number of overshot water wheels were in operation in the last century,63

only few series of tests were performed. Most of the test results were never64

published in hydraulic engineering textbooks or journals and they are only65

available in not widely known reports and articles [4] [7] [15] [16] [17].66

Therefore, scope of this work is to develop a theoretical model in order67

to calculate the different kinds of power losses occurring inside an overshot68

water wheel. The theoretical results are validated on experimental ones and69

the experimental characteristic curves are illustrated.70

4

2. Theory71

Fig. 1 shows an overshot water wheel with an internal radius R, an72

angular velocity ω and an angular distance between two blades β = 2π/nb73

where nb is the total number of the blades. The angular position of each74

blade is φ, where φ = 0 is the vertical line passing for the rotation axle.75

The water jet enters into each cell at its natural angle of fall, as a fast76

and thin sheet; the opening of each bucket is slightly wider than the jet,77

to let the air escape. However, a portion of the inflowing water is lost,78

generating a power loss LQu . The jet fills into the buckets impinging on the79

blades, dissipating a portion of its kinetic energy and generating the impact80

power loss Limp. The water volume in the bucket moves then with it at the81

mean tangential velocity u = ω·(R+hs/2), where hs is the width of the82

bucket along the wheel radius. After a rotation of φ = φe from the starting83

position (the top of the wheel), the water begins to spill out, generating84

volumetric losses LQr . When the bucket reaches the lowest point of the85

wheel (φ = π) the emptying process completes.86

In the present analysis, we suppose the water inside the cells to be at87

rest and the effect of the centrifugal force, which makes its surface profile88

not to be horizontal, is taken into account.89

The total input power P the wheel disposes depends on the hydraulic90

and geometric boundary conditions and it is expressible by (Fig. 1):91

P = Pu−Pd = γ ·Q·(Hu−Hd) = γ ·Q·∆H = γ ·Q·(

∆Hg +~vu

2 − ~vd2

2g

)(1)

where Pu is the power of the stream before the wheel and Pd that down-92

5

stream the wheel, γ = 9810 N/m3 the water specific weight and Q is the93

incoming flowrate. ∆H = Hu−Hd is the net head, ∆Hg the geometric dis-94

tance between the free surfaces, ~vu and ~vd the upstream and downstream95

water velocity, respectively, and g = 9.81 m/s2 is the gravity acceleration.96

In our case ~vd ' 0 and the free surface of the tailrace is located just under97

the bottom of the wheel.98

The mechanical output power Pout is expressible by:99

Pout = P −∑

Losses = P − Limp − Lt − Lg − LQu − LQr (2)

where Limp is the power loss occurring in the impact, Lt the impact loss gen-100

erated when the blades impact on the tailrace (if the blades are submerged101

in the tailrace), Lg the mechanical friction loss at the shaft supports, LQu102

the volumetric loss at the top of the wheel and LQr the volumetric loss103

during rotation. The amount of input power which is used for producing104

useful work is expressed by the efficiency η:105

η =PoutP

=P −

∑Losses

P= 1−

∑Losses

P(3)

106

In the next sections a detailed analysis of the power losses is reported.107

2.1. Impact losses108

Impact losses may occur both at the top of the wheel and at the tailrace.109

The former is generated at the top of the wheel when the jet enters into110

the cells. Called ~v the absolute velocity of the water and ~u the tangential111

6

velocity of the blades in the impact point, the relative velocity of the jet is112

~w = ~v − ~u. The impact power loss can be written as:113

Limp = ξγQw2

2g(4)

where ξ is the impact coefficient and |~v| =√|~vu|2 + 2g(hu + hs/2) is the114

intensity of the jet absolute velocity; hu is the water depth in the conveying115

channel and hs the depth of the bucket opening.116

In general, the impact torque also contributes to the power generation.117

This contribution decreases with the wheel angular speed (due to the reduc-118

tion in the relative velocity) and disappears once the jet velocity reaches the119

velocity of the cell. However, during rotation, the jet impinges generally on120

the external surface of the blades, due to their backward inclination, or on121

the shroud of the wheel. Therefore it is reasonable to assume ξ = 1, since122

the jet cannot efficiently perform additional work flowing along the blades.123

Moreover, the impact torque contribution to the power generation is related124

to the stream kinetic term, which is much lower than the potential one (fur-125

ther, the exchanged kinetic power depends on the relative entry velocity,126

and not directly on the absolute jet velocity). A different reasonable value127

of ξ does not affect appreciably the results. In [7], the effect of the entry128

velocity has been investigated, changing the conveying channel slope; for129

high entry velocity, the impact torque becomes important. However, in130

order to reach higher efficiency, it would be more advisable to reduce the131

channel slope (thus the entry velocity of the jet) and increase the geometric132

head ∆Hg, thus the wheel diameter, since the potential energy of water is133

better exploited than the kinetic one.134

7

The impact loss at the tailrace (Lt), occurs if the blades near the low-135

ermost position (φ = π) are submerged in the tailrace. The loss Lt can be136

expressed as:137

Lt =1

2gCDγ(~u− ~vd)

2A~u (5)

where ~vd is the water velocity at the tailrace under the wheel, CD the drag138

coefficient (depending on the shape of the blade) and A the area of the139

blade interested to the impact.140

2.2. Mechanical losses141

The only mechanical power loss occurring in the investigated wheel is142

due to the friction at the shaft supports. It can be expressed by:143

Lg = M · ω = W · f · r · ω = (Wwh +Wwat) · f · r · ω (6)

where M is the resistance torque due to the friction at the shaft supports,144

ω the angular velocity of the wheel, W = Wwh + Wwat the total weight of145

the wheel, where Wwat is the weight of the water in the buckets and Wwh is146

the weight of the wheel, f the friction coefficient ('1/16 between lubricated147

steel surfaces) and r the shaft level arm.148

2.3. Volumetric losses149

Volumetric losses occur both at the top of the wheel (LQu) and during150

rotation (LQr).151

A portion of the incoming flowrate is lost at the top of the wheel, de-152

termining a power loss LQu :153

8

LQu = γ ·Qu · (Hu −Hd) (7)

where Qu is the flowrate which cannot enter into the buckets and it is a154

function of the wheel velocity and the blades shape. Due to the complex155

filling and impact process, Qu is difficult to estimate theoretically. We call156

χ = LQu/P = Qu/Q and it will be estimated by a numerical optimization157

process.158

Volumetric losses LQr occur when there is a quantity of water Qr spilling159

out the buckets during wheel rotation. Since the angular distance between160

two buckets is β, LQr has a periodic cycle of T = β/ω. The instantaneous161

power loss at time t = tj is:162

LQr(t = tj) = γn∑i

Qr,i · (Hw,i −Hd) (8)

where Qr,i is the flowrate exiting from each bucket i at time t = tj, which163

energy head is Hw,i, and n is the number of the buckets contributing to the164

outflow.165

Qr,i is expressible by:166

Qr,i =V (φi, ω)− V (φi + dφ, ω)

dφ/ω= −ω∂V

∂φ

∣∣φ=φi

(9)

where V (φi, ω) is the maximum water volume each bucket i can contain in167

its position φ = φi for a specific angular velocity ω. The average value of168

LQr can be expresses by:169

LQr =γ∫ t+Tt

∑ni Qr,i · (Hw,i −Hd)

T(10)

9

In order to avoid an early loss of water, each cell should only be filled170

with 30 - 50% of its volume, so that the outflow starts at a very low level171

[4].172

The spilling out process from a bucket starts at position φ = φe where173

Vin = V (φ, ω), with Vin the water volume entered into the bucket and174

V = V (φ, ω) the maximum water volume the bucket can contain (Fig. 1);175

all the buckets beyond the position φe contribute to LQr .176

The volume Vin is expressible by:177

Vin = (Q−Qu) · β/ω (11)

while V = V (φ, ω) depends on the geometric shape of the blades and it178

decreases with the angular position φ. The volume V decreases also with179

the rotational speed of the wheel, since the centrifugal force, acting on the180

water volume, inclines the water surface profile, as depicted in Fig.1.181

For the theoretical model, two different cases are here considered: the182

former is a simplified case, where the effect of the centrifugal force is ne-183

glected, while in the second one the centrifugal force is taken into account.184

If the centrifugal force is neglected and the water in the buckets is sup-185

posed to be at rest, the water surface profile is horizontal; in this simplified186

case, the maximum water volume is called Vs = Vs(φ) and it depends only187

on the shape of the buckets and on their position φ. Being independent to188

ω, Vs = Vs(φ) is a characteristic parameter of each wheel and Fig. 2 depicts189

it for the installed wheel. The volume Vs reduces drastically after a rotation190

of φ ' π/2 rad from the top of the wheel (where the water fills into the191

bucket and φ = 0). Instead, if the effect of the centrifugal force is consid-192

10

ered, the maximum water volume is V = V (φ, ω) < Vs(φ), since the water193

surface profile inside the buckets is inclined and not horizontal. The water194

surface in each bucket disposes in order to make its profile in each point195

perpendicular to the total acting force ~Ft = ~Fc + ~Fg, where ~Fc = ω2r196

is the centrifugal acceleration (force for unit mass) and ~Fg = ~g the gravity197

acceleration, with r the distance between each point of the water surface in198

the buckets and the wheel rotation axle. In this case the maximum water199

volumes the buckets can contain are lower than those of the simplified case.200

For the sake of simplicity, assuming the water surface profile to be linear201

and the mean distance r = r, V can be calculated assuming that Vs reduces202

of a quantity of about Vl ' 1/2bho2 sinα: ho = ho(φ) is the maximum hor-203

izontal length of the water surface inside the buckets, α = α(φ, ω) is the204

inclination of the water surface due to the centrifugal effect (in the point205

which distance is r from the axle) and b the width of the buckets (Fig. 1).206

The higher ω the higher α and the reduction in the contained volume, so207

that V = Vs − Vl = V (φ, ω).208

3. Method209

A 1:2 steel model of an overshot water wheel was installed in the Labo-210

ratory of Hydraulics at Politecnico di Torino (Fig. 3) and 256 operative con-211

ditions were investigated, by varying the entry flowrate (Q = 0.01 ÷ 0.137212

m3/s) and the wheel rotation speed (ω = 0.5÷3.5 rads/s). The water depth213

hu at the end of the conveying channel varied from 0.02 m to 0.09 m, and214

the stream velocity |~vu| varied between 0.5 and 1.5 m/s, depending on the215

flowrate. Therefore, the entry jet velocity |~v| ranged between 1.5 and 2.4216

11

m/s.217

The wheel has a maximum diameter of D=1.4 m and a width of b=1 m,218

while the conveying channel has a width of bc=0.96 m, narrower than the219

wheel to allow for ventilation. The number of the curved blades, fastened220

to the lateral shrouds, is 24 and the weight of the wheel is Wwh ' 4300 N.221

The depth of the cells is hs = 0.157 m.222

The total discharge Q was imposed acting on a pump and detected223

by an electromagnetic flow meter, which accuracy was ±0.5 · 10−3 m3/s.224

The water level hu just upstream the wheel was measured by an ultrasonic225

sensor, with an accuracy of ±0.002 m. Starting from the water level hu and226

the flowrate Q, the stream velocity vu = Q/(bchu) and the total head Hu of227

the stream could be determined (see eq. 1).228

The wheel rotation speed ω was regulated by an electromagnetic brake,229

constituted by a generator and a resistor. The measurement of the wheel an-230

gular velocity was very accurate, measured by the acquisition board internal231

clock, which could discretize the proximity sensor output signal frequency232

till 100 MHz. Before the brake, a gearbox amplified the low wheel rotation233

speed of 15.6 times.234

A torque transducer registered the output torque C at the shaft axle.235

By multiplying the applied torque for the angular velocity, the output ex-236

perimental power Pexp = C · ω could be determined and compared to the237

theoretical one Pout. The power accuracy depends mainly on the torque238

transducer accuracy which is ±6 Nm.239

Scope of the work is the determination of the performance characteristics240

of overshot water wheels and the quantification of the different kinds of241

12

power losses occurring in them; in particular, the estimation through an242

optimization process of the volumetric losses LQu are carried out.243

4. Results and discussion244

4.1. Experimental results245

Fig. 4, Fig. 5a and Fig. 5b show the experimental results for the mechan-246

ical power and efficiency. First of all, two different limit rotational speeds247

can be identified: the runaway velocity ωr, in correspondence to which the248

output power tends to become null (ωr ' 4 ÷ 4.5 rad/s) and the critical249

velocity ωcr, when the output power and efficiency begin to decrease sharply250

(ωcr ' 2.7 rad/s or vcr = 1.8 m/s). However, the critical velocity can only251

be observed for flowrates bigger than 0.03 m3/s, to which correspond entry252

velocity of the jet 1.8 < v < 2.4 m/s, or 0.75 < vcr/v < 1 (the maximum253

investigated flowrate was 0.137 m3/s, with an entry absolute jet velocity of254

2.4 m/s). Therefore, the power losses increase dramatically and the effi-255

ciency reduces once the cell speed approaches 75% of the inflow velocity. In256

Fig. 4 it can be observed that the output power decreases with the increase257

in the rotation speed of the wheel; for higher Q and ω, most of the water258

cannot fill into the buckets, it slips around the external part of the blades259

and LQu increases (as illustrated in Fig. 1). Fig. 5a shows a constant effi-260

ciency of ' 80% with the angular wheel speed, for a wide range of flowrates261

(0.01-0.07 m3/s). In this range the optimal filling ratio (the water volume262

to the bucket volume) is included between 0.23 to 0.5. The efficiency tends263

to decrease for ω > ωcr = 2.7 rad/s as a consequence of the increase in264

LQu mainly. Fig. 5b depicts the maximum efficiency versus the dimension-265

13

less flowrate, where Qmax = 0.05 m3/s is the flowrate corresponding to the266

maximum efficiency. Fig. 5b clearly shows that overshot wheels are very267

suitable hydraulic machines in sites with small and variable water flow, with268

constant efficiency for a wide range of flowrates. The maximum efficiency269

is 85% for Q = 0.05 m3/s and then it decreases, due to the increase in the270

volumetric losses, mainly those at the top of the wheel.271

In [4] a complete and detailed review of old work for overshot water272

wheels is reported; similar curves of those shown in Fig. 5 are illustrated,273

where the efficiency begins to reduce when the cell speed approaches 80%274

of the inflow velocity. Therefore, the uppermost wheel tangential velocity275

should be about 0.8 of the jet absolute velocity. Moreover, [4] shows con-276

stant efficiency η > 80% for a wide range of flowrates, up to the flowrate277

corresponding to the maximum efficiency Qmax. The efficiency trend for278

flowrates higher than Qmax is not investigated, thus our experimental tests279

can be very useful to complete and detail the behavior of overshot water280

wheels.281

4.2. Theoretical results282

Fig. 6 depicts the theoretical power versus the experimental one, as-283

suming Qu = 0 and LQu = 0. It is clear that for low discharges (Q < 0.04284

m3/s) the hypothesis for the calculation of Limp, Lg and LQr are well posed,285

since theoretical powers are aligned along the bisecting line. In these cases286

the entirety of the discharge enters into the buckets and after a little rota-287

tion the water can be considered at rest, as confirmed in the experimental288

tests. For Q > 0.04 m3/s, in the cases with high rotation velocity (to which289

correspond lower values of Pexp) Pout disposes over the bisecting line, be-290

14

ing overestimated with respect to Pexp. It is reasonable to attribute the291

overestimation to the volumetric losses LQu and to calculate the unknown292

Qu by an optimization process. The best value of Qu for each case is that293

minimize the difference between the theoretical power Pout and the exper-294

imental one Pexp. After that a value for the outflow Qu is assumed, the295

related power loss LQu can be calculated by eq. 7 and, when the position φe296

where Vin = V (φ, ω) is deducted, the outflow Qr from each of the n buckets297

at φ > φe can be calculated by eq. 9. Then, by eq. 8 and eq. 10 it is possi-298

ble to evaluate LQr and by eq. 6 to calculate Lg, being known Wwat. Pout299

can be then calculated by eq. 2. Instead, for the lowest ω (or the biggest300

Pexp) the theoretical output power Pout is lower than the experimental one;301

it happens because of the more pronounced oscillation motion inside the302

buckets, not considered in this paper, caused by the higher values of the303

relative jet velocity.304

From the theoretical results it is resulted that the bigger Q the bigger305

the power losses, while their dependency to the rotational velocity ω is less306

intuitive. The volumetric losses LQr and LQu increase with Q because the307

spilling out happens early and the portion of Q which cannot fill into the308

buckets enhances. The impact loss Limp enhances as a consequence of the309

increase in the jet velocity and flowrate, and friction losses Lg increase as a310

result of the increase in the water weight in the buckets. The tailrace loss311

is considered to be Lt = 0, since the wheel is uplift on the tailrace.312

Fig. 7 shows the trend of χ = LQu/P with ω; for ω > ωcr ' 2.7 rad/s,313

χ increases linearly. The trend equation for χ is:314

315

15

χ =LQu

P' 0 for ω < ωcr

χ =LQu

P= 2.2 ·

[ωωcr− 1]

for ω > ωcr

(12)

316

Fig. 8 depicts the trend of LQr/P . For ω < ωcr the bigger the rotational317

speed the bigger LQr ; although the water volume Vin which fills into the bu-318

ckets decreases with ω, the increasing effect of the centrifugal force becomes319

more important and the spilling out begins earlier. The maximum value of320

the water surface inclination inside the buckets due to the centrifugal effect321

is α ' 40◦. Then, at ω = ωcr, LQr/P begins to decrease sharply, due to the322

fact that the rapid enhancement in LQu reduces considerably Vin. For very323

low angular velocity the entirety of the discharge fills into the bucket and324

Vin increases, but the effect of the centrifugal force reduces. For ω → 0 the325

buckets fill completely and the emptying process starts immediately.326

The trend equation of LQr/P is:327

328

LQr

P= 0.24 ·

(ωωcr

)2

− 0.01 · ωωcr

+ 0.04 for ω < ωcr

LQr

P= −1.6 ·

(ωωcr

)2

+ 2.9 · ωωcr− 1.0 for ω > ωcr

(13)

329

Fig. 9 depicts the trend of Limp/P versus ω/ωcr; Limp/P increases with330

ω, but for ω > ωcr it decreases, because the enhancement in LQu reduces the331

flowrate impacting on the wheel. For ω → 0 the entirety of the discharge332

fills into the bucket and u → 0. The relative velocity becomes ~w = ~v333

and Limp/P = vu2/2g+hu+hs/2∆H

, where vu = vu(Q), hu = hu(Q) and ∆H =334

∆H(Q) (because ∆H = ∆H(vu, hu)). In correspondence of ω/ωcr = 1 there335

16

is the maximum, which value is 12%, lower than that for LQr/P = 32%.336

337

The trend equation for Limp/P is:338

Limp

P= 0.04 ·

(ωωcr

)2

− 0.002 · ωωcr

+ 0.06 for ω < ωcr

Limp

P= −0.39 ·

(ωωr

)2

+ 0.71 · ωωr− 0.20 for ω > ωcr

(14)

339

Fig. 10 shows the increase of Lg/P with ω. Although Wwat in eq. 6340

decreases with ω, since Wwat is negligible than the weight of the wheel341

Wwh, Lg/P = M · ω/P increases with ω. For ω → 0 Lg tends to 0. The342

trend equation for Lg/P is:343

LgP

= 0.027 ·(

Q

Qmax

)−1

·(ω

ωcr

)(15)

depending on the wheel velocity ω and the discharge Q.344

Fig. 11 depicts the theoretical powers calculated by eq. 2 versus the345

measured ones and the results are well aligned along the bisecting line. For346

low discharges and Pexp < 800 W the model predicts the output power very347

well and the highest errors are related to high discharges (corresponding to348

the lowest efficiency); it is a satisfactory result, since overshot water wheels349

are used in sites with low discharges.350

The global average error among the experimental powers (Pexp) and the351

theoretical ones (Pout) can be expressed as:352

er =1

ne

ne∑j=1

erj =1

ne

ne∑j=1

|Poutj − Pexpj |Pexpj

(16)

17

where ne is the total number of the experiments and Pout is calculated by353

eq.2. Expressing the volumetric losses LQu by eq.12, er = 8.2%.354

Imposing the efficiency η in eq. 3 equal to zero and substituting in it355

eq. 12, 13, 14, 15 for ω > ωcr, it is possible to obtain the value of ω356

corresponding to the runaway velocity ωr, for Q ≥ 0.03 m3/s, as a function357

of the discharge.358

P − Limp − Lg − LQu − LQr

P=

= a−(ωrωcr

)·[(b+

c

(Q/Qmax)

)− d

(ωrωcr

)]= 0 (17)

In our case a = 4.44, b = 5.88, c = 0.027 and d = 2.02, ωcr = 2.7 rad/s359

and Qmax = 0.05 m3/s the maximum efficiency flowrate. The obtained360

solution ω corresponds to ωr, when the total extracted power from the361

stream is dissipated by friction and other losses. Eq. 17 can be solved362

analytically. The runaway velocity ωr depends slightly on the flowrate; ωr =363

4.2÷4.3 rad/s as confirmed by the experimental results (Fig. 4 and Fig. 5a).364

Fig. 12 depicts the calculated runaway velocity versus the discharge.365

Tab. 1 depicts the minimum and maximum values for each power loss,366

as a percentage of the input power. LQu/P represents the most important367

loss for ω > ωcr, while LQr/P and Limp/P have a maximum of 32% and368

12% at ω = ωcr, respectively; Lg is the smallest one. All the power losses369

depend strictly on the rotational speed ω. The bigger ω the bigger LQu and370

Lg, while for LQr and Limp there is a maximum at ω = ωcr ' 2.7 rad/s.371

The increase in the discharge Q makes all the power losses enhance.372

In order to reduce the power losses and to increase the performance of373

18

the wheel, it is necessary to reduce mainly LQu , for example by new recovery374

systems [19], or by the improvement of the blades geometry.375

19

5. Conclusion and further works376

Water wheels have been used for hundred years for the production of377

energy, but with the development of turbines, steam engines and electric378

motors their employment ceased. Since their sustainability, low costs and379

high efficiency, they still represent a suitable technology to product energy380

in site with low heads.381

In particular, since the efficiency of overshot wheels remains high for a382

wide range of flowrates, they become suitable also in sites where the water383

resource is variable in time or lacking. However, when the flowrate and384

rotational speed exceed certain values, volumetric losses determine a waste385

of water, so a decrease in efficiency and possible damages to the wheel.386

Therefore, experimental analyses and theoretical models are necessary to387

understand the optimal operative conditions for the wheel, both for engi-388

neering design and for scientific purposes.389

Thank to our analyses and a literature review, it is possible to say that,390

in general, each cell should only be filled with up to 30–50% of its volume,391

in order to reduce volumetric losses during rotation. The buckets should be392

shaped in a way so that the water jet can enter each cell at its natural angle393

of fall, with an opening of each cell slightly wider than the jet, to let the394

air escape. However, in the investigated overshot wheel, it is the potential395

energy of water (the water weight) which constitutes the main torque for396

the wheel. The shape of the cells should retain the water inside of the cell397

until the lowermost position, when it finally empties rapidly. The wheel398

diameter is determined by the head difference, while the relative velocity is399

not a simple parameter to optimize. When the wheel rotates with a very400

20

low speed, the cells fill completely. When the cells move too fast, only401

little water can enter each cell, since volumetric losses. From experimental402

results we can claim that for wheel tangential velocity lower than 75-80%403

of the entry water velocity, the efficiency is high and quite constant with404

the wheel velocity (see sec.4.1; for further details see [4]).405

In our study case, two different limit rotational speeds ω are identified:406

the runaway velocity ωr, in correspondence to which the output power tends407

to become null (ωr ' 4.2÷4.3 rad/s) and the critical velocity ωcr, where the408

output power begins to decrease brusquely (ωcr ' 2.7 rad/s). For ω < ωcr409

the efficiency is higher than 80% for a wide range of flowrates.410

For the future, CFD simulation will be used to investigate the complete411

and complex hydraulic behavior of the water inside the wheel. Indeed, for412

low angular speeds, the high values of the relative velocity ~w make the413

assumption of the resting water inside the cells not to be totally true; the414

oscillation motion of the water inside the buckets affects the position and415

inclination of the water surface. CFD simulations will be also useful to416

investigate other hydraulic conditions and geometries. The future main417

goal will be the improvement of the efficiency also for high discharges and418

rotational speeds, mainly reducing volumetric losses.419

6. Acknowledgements420

The research leading to these results has received funding from Orme421

(Energy optimization of traditional water wheels – Granted by Regione422

Piemonte via the ERDF 2007-2013 – Partners Gatta srl, BCE srl, Rigamonti423

Ghisa srl, Promec Elettronica srl and Politecnico di Torino).424

21

7. List of variables425

α = water surface inclination inside the buckets [rad]426

β = angular distance between two buckets [rad]427

∆H = difference of energy head [m]428

∆Hg = vertical distance between free surfaces [m]429

η = wheel efficiency430

φ = angular coordinate [rad]431

φe = position at which the emptying process begins [rad]432

ω = angular wheel velocity [rad/s]433

ωcr = critical angular velocity [rad/s]434

ωr = runaway velocity [rad/s]435

γ = water specific weight [N/m3]436

ξ = impact coefficient437

χ = dimensionless volumetric loss at the top of the wheel438

A = area of the blade submerged in the tailrace [m2]439

b = wheel width [m]440

bc = channel width [m]441

C = shaft measured torque [Nm]442

CD = drag coefficient443

f = friction coefficient at the shaft support444

Fc = centrifugal force for unit mass [m/s2]445

Fg = gravity force for unit mass [m/s2]446

Ft = total acting force on the water volume inside the buckets for unit447

mass [m/s2]448

g = gravity acceleration [m/s2]449

22

ho = bucket depth projected on the horizontal [m]450

hs = bucket depth [m]451

hu = upstream water depth [m]452

Hd = downstream energy head [m]453

Hu = upstream energy head [m]454

i = general bucket455

Lg = friction power loss [W]456

Limp = impact power loss [W]457

LQr = volumetric power loss during rotation [W]458

LQu = volumetric power loss at the top of the wheel [W]459

Lt = tailrace impact loss [W]460

n = number of buckets contributing to the outflowing process461

nb = number of blades462

ne = number of experimental tests463

P = available input power [W]464

Pexp = experimental output power [W]465

Pout = theoretical output power [W]466

Pd = downstream power [W]467

Pu = upstream power [W]468

Q = incoming flowrate [m3/s]469

Qr = volumetric loss during rotation [m3/s]470

Qu = volumetric loss at the top of the wheel [m3/s]471

r = shaft level arm [m]472

R = internal wheel radius [m]473

t = general time step [s]474

23

T = periodic cycle of the power generation [s]475

u = wheel tangential speed [m/s]476

vcr = critical tangential velocity [m/s]477

vd = downstream water velocity [m/s]478

vu = upstream water velocity [m/s]479

V = maximum water volume for a moving bucket [m3]480

Vl = reduction of the water volume in the bucket [m3]481

Vs = maximum water volume for a bucket at rest [m3]482

w = entry relative velocity [m/s]483

W = wheel plus water weight [N/m3]484

Wwh = wheel weight [N/m3]485

Wwat = water weight inside the buckets [N/m3]486

8. Bibliography487

[1] International Energy Agency, World energy outlook, 2013.488

[2] D. Capecchi, Over and undershot waterwheels in the 18th century.489

Science-technology controversy, Advances in Historical Studies, 2013;490

2(3): 131-139.491

[3] J. A. Senior, Hydrostatic pressure converters for the exploitation492

of very low head hydropower potential, Ph.D. thesis, University of493

Southampton (2009).494

[4] G. Muller, K. Kauppert, Performance characteristics of water wheels,495

Journal of Hydraulic Research, 2004; 42(5): 451-460.496

24

[5] G. Muller, S. Denchfield, R. Marth, R. Shelmerdine, Stream wheels497

for applications in shallow and deep water, 2007, Proc. 32nd IAHR498

Congress, Venice C (2c, Paper 291).499

[6] M. Denny, The efficiency of overshot and undershot waterwheels, Eu-500

ropean Journal of Physics, 2004; 25(2): 193-202.501

[7] A. Williams, P. Bromley, New ideas for old technology-experiments502

with an overshot waterwheel and implications for the drive system,503

International Conference of Gearing, transmission and mechanical sys-504

tem, 2000, pages 781-790.505

[8] G. Muller, C. Wolter, The breastshot waterwheel: design and model506

tests, Proceedings of the ICE-Engineering Sustainability, 2004; 157(4):507

203-211.508

[9] C. Bach, Die Wasserrader: Atlas (The water wheels: technical draw-509

ings, 1886, Konrad Wittwer Verlag, Stuttgart (in German).510

[10] E. Garuffa, Macchine motrici ed operatrici a fluido, 1897, Meccanica511

industriale, published by U. Hoepli (in Italian).512

[11] F. Chaudy, Machines hydrauliques, Bibliotheque du conducteur de513

travaux publics, 1896, published by Vve. C. Dunod et P. Vicq (in514

French).515

[12] I. Church, Hydraulic Motors, with Related Subjects, Including Cen-516

trifugal Pumps, Pipes, and Open Channels, Designed as a Text-book517

for Engineering Schools, published by J. Wiley & sons, 1914.518

25

[13] A. Morin, E. Morris, Experiments on Water wheels Having a Vertical519

Axis, Called Turbines. Published at Metz and Paris, Franklin Institute,520

1843.521

[14] J. Poncelet, Memoria sulle ruote idrauliche a pale curve, mosse di sotto,522

seguita da sperienze sugli effetti meccanici di tali ruote. Translated by523

Errico Dombre, Dalla tipografia Flautina, Napoli, 1843.524

[15] C. Weidner, Theory and test of an overshot water wheel, Bulletin of525

the University of Wisconsin No. 529, Engineering Series, 1913, 7(2),526

117–254.527

[16] C. Weidner, Test of a steel overshot water wheel, Engng. News, 1913,528

No. 69(1), 39–41.529

[17] K. Meerwarth, Experimentelle und theoretische untersuchungen am530

oberschlachtigen wasserrad. (experimental and theoretical investiga-531

tion of an overshot water wheel), Ph.D. thesis, Technical University of532

Stuttgart/Germany (in German) (1935).533

[18] E. Quaranta, R. Revelli, Performance characteristics, mechanical out-534

put power and power losses estimation for a breastshot water wheel,535

International Journal of Energy, 2015, In press.536

[19] B. Wahyudi, A. Faizin, S. Suparman, Increasing efficiency of overshot537

waterwheel with overflow keeper double nozzle, Applied mechanics and538

materials, 2013, 330: 209-213.539

26

List of Figures540

1 Reference scheme for the overshot water wheel. The energy541

line is called E.l., Q is the total flowrate and Hu and Hd are542

the energy heads upstream and downstream the wheel. Hw,i543

is the energy head of the discharge Qr which spills out the544

generic bucket i, with LQr the volumetric loss during rota-545

tion. Qu is the flowrate loss at the top of the wheel and LQu546

the related power loss. Limp is the hydraulic impact loss and547

Lg the mechanical friction loss at the shaft. vu is the up-548

stream velocity and vd the downstream one. The gray areas549

represent the water volumes contained in the buckets, which550

water surface is inclined of α = α(φ, ω) as a consequence of551

the centrifugal force; in position φ = φe = φe(Q,ω) the water552

begins to spill out. The arrows represent the water flows, in553

particular the discharge Q and the discharge losses Qr and Qu. 1554

2 The maximum water volume Vs = Vs(φ) the bucket can con-555

tain for the installed wheel. . . . . . . . . . . . . . . . . . . 2556

3 Lateral view of the experimental water wheel. The torque557

transducer, the gearbox (blue box) and the generator (black558

cylinder) are assembled at the shaft. The wheel has an ex-559

ternal diameter of 1.4 m. . . . . . . . . . . . . . . . . . . . . 3560

4 Output power Pexp for different flowrates Q and wheel rota-561

tion speeds ω. The output power decreases with the rota-562

tional speed and increases with the flowrate. . . . . . . . . 4563

27

5 (a) The efficiency η versus the rotation speed ω, for differ-564

ent flowrates Q. The critical rotational speed is ωcr = 2.7565

rad/s, for which the output power and efficiency begin to566

decrease rapidly. (b) The maximum efficiency versus the di-567

mensionless flowrate Q/Qmax, where Qmax = 0.05 m3/s is568

the maximum efficiency flowrate. . . . . . . . . . . . . . . . 5569

6 Theoretical power results (Pout) versus experimental ones570

(Pexp) for different flowrates Q, assuming LQu = 0; it is a571

very good assumption for small flowrates (Q <0.04 m3/s). . 6572

7 Dimensionless volumetric loss χ = LQu/P versus the dimen-573

sionless angular speed of the wheel ω/ωcr, for different di-574

scharges Q; LQu is the volumetric loss due to the water which575

cannot fill into the buckets, P the available input power and576

ωcr = 2.7 rad/s the critical velocity. . . . . . . . . . . . . . 7577

8 Dimensionless volumetric loss LQr/P versus the dimension-578

less angular speed of the wheel ω/ωcr, for different discharges579

Q; LQr is the volumetric loss from the buckets during rota-580

tion, P the available input power and ωcr = 2.7 rad/s the581

critical velocity of the wheel. . . . . . . . . . . . . . . . . . . 8582

9 Dimensionless impact loss Limp/P versus the dimensionless583

angular speed of the wheel ω/ωcr, for different discharges584

Q. Limp is the impact loss, P the available input power and585

ωcr = 2.7 rad/s the critical velocity of the wheel. . . . . . . . 9586

28

10 Dimensionless friction loss Lg/P versus dimensionless angu-587

lar speed of the wheel ω/ωcr, for different discharges Q. Lg588

is the friction loss, P the available input power and ωcr = 2.7589

rad/s the critical velocity of the wheel. . . . . . . . . . . . . 10590

11 Analytical power Pout versus experimental one Pexp for dif-591

ferent discharges Q. The average error is 8.2%. . . . . . . . . 11592

12 Runaway velocity ωr versus the dimensionless flowrateQ/Qmax,593

where Qmax = 0.05 m3/s (eq.17). The runaway velocity ωr594

remains constant at about ωr = 4.3 rad/s. . . . . . . . . . . 12595

29

Figure 1: Reference scheme for the overshot water wheel. The energy line

is called E.l., Q is the total flowrate and Hu and Hd are the energy heads

upstream and downstream the wheel. Hw,i is the energy head of the dis-

charge Qr which spills out the generic bucket i, with LQr the volumetric

loss during rotation. Qu is the flowrate loss at the top of the wheel and

LQu the related power loss. Limp is the hydraulic impact loss and Lg the

mechanical friction loss at the shaft. vu is the upstream velocity and vd the

downstream one. The gray areas represent the water volumes contained in

the buckets, which water surface is inclined of α = α(φ, ω) as a consequence

of the centrifugal force; in position φ = φe = φe(Q,ω) the water begins to

spill out. The arrows represent the water flows, in particular the discharge

Q and the discharge losses Qr and Qu.

Figure 2: The maximum water volume Vs = Vs(φ) the bucket can contain

for the installed wheel.

2

Figure 3: Lateral view of the experimental water wheel. The torque trans-

ducer, the gearbox (blue box) and the generator (black cylinder) are assem-

bled at the shaft. The wheel has an external diameter of 1.4 m.

3

Figure 4: Output power Pexp for different flowrates Q and wheel rotation

speeds ω. The output power decreases with the rotational speed and in-

creases with the flowrate.

4

(a) a. (b) b.

Figure 5: (a) The efficiency η versus the rotation speed ω, for different

flowrates Q. The critical rotational speed is ωcr = 2.7 rad/s, for which the

output power and efficiency begin to decrease rapidly. (b) The maximum

efficiency versus the dimensionless flowrate Q/Qmax, where Qmax = 0.05

m3/s is the maximum efficiency flowrate.

5

Figure 6: Theoretical power results (Pout) versus experimental ones (Pexp)

for different flowrates Q, assuming LQu = 0; it is a very good assumption

for small flowrates (Q <0.04 m3/s).

6

Figure 7: Dimensionless volumetric loss χ = LQu/P versus the dimension-

less angular speed of the wheel ω/ωcr, for different discharges Q; LQu is the

volumetric loss due to the water which cannot fill into the buckets, P the

available input power and ωcr = 2.7 rad/s the critical velocity.

7

Figure 8: Dimensionless volumetric loss LQr/P versus the dimensionless

angular speed of the wheel ω/ωcr, for different discharges Q; LQr is the

volumetric loss from the buckets during rotation, P the available input

power and ωcr = 2.7 rad/s the critical velocity of the wheel.

8

Figure 9: Dimensionless impact loss Limp/P versus the dimensionless angu-

lar speed of the wheel ω/ωcr, for different discharges Q. Limp is the impact

loss, P the available input power and ωcr = 2.7 rad/s the critical velocity

of the wheel.

9

Figure 10: Dimensionless friction loss Lg/P versus dimensionless angular

speed of the wheel ω/ωcr, for different discharges Q. Lg is the friction loss,

P the available input power and ωcr = 2.7 rad/s the critical velocity of the

wheel.

10

Figure 11: Analytical power Pout versus experimental one Pexp for different

discharges Q. The average error is 8.2%.

11

Figure 12: Runaway velocity ωr versus the dimensionless flowrate Q/Qmax,

where Qmax = 0.05 m3/s (eq.17). The runaway velocity ωr remains constant

at about ωr = 4.3 rad/s.

12

Table 1: Minimum and maximum value of the dimensionless power losses.

L/P Limp/P LQu/P LQr/P Lg/P

Min [%] 6 0 0.8 1.2

Max [%] 12 76 32 7

13