Hydraulic Behavior and Performance of Breastshot Water Wheels for Different Numbers of Blades PDF

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Hydraulic Behavior and Performance of Breastshot Water Wheels for Different Numbers of Blades Emanuele Quaranta 1 and Roberto Revelli 2 Abstract: Thanks to their efficiency and simplicity of contruction, breastshot water wheels represent an attractive low head hydropower converter. In this work, a b...

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Hydraulic Behavior and Performance of Breastshot Water Wheels for Different Numbers of Blades

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Emanuele Quaranta 1 and Roberto Revelli 2

Abstract: Thanks to their efficiency and simplicity of contruction, breastshot water wheels represent an attractive low head hydropower converter. In this work, a breastshot wheel is investigated by numerical simulations, and the results are validated with experimental tests. The average discrepancy between the numerical shaft torque and the experimental torque is lower than 5%. The numerical model is then used to investigate the performance and the hydraulic behavior of the wheel for different numbers of blades (16, 32, 48, and 64 blades) and at different hydraulic conditions. The increase in efficiency from 16 blades to the optimal blades number ranges between 12 to 16% in function of the hydraulic conditions. Empirical laws are also reported to quantify the improvement in efficiency with the blades number. These laws can support the design process of similar breastshot water wheels. The optimal blades number for this kind of wheel is identified in 48. DOI: 10.1061/(ASCE)HY.1943-7900.0001229. © 2016 American Society of Civil Engineers. Author keywords: Blades number; Computational fluid dynamics (CFD); Hydropower; Mini-hydro; Water wheels.

Introduction In the European Commission legislations, electricity production in large scale from renewable energy sources has become a major purpose for meeting the important renewable energy targets and to limit greenhouse gas emissions. As a consequence, a new and wide interest on renewable sources is spreading in Europe and also worldwide, especially the energy production by wind, solar, and hydro power. With respect to wind and solar energy production, hydroelectricity generally exhibits some advantages. Hydro plants are more responsive to load management requirements (they can be managed by human control easier), and hydro output is more predictable than solar and wind output. The drawback of wind and solar resources is their variability in time, which introduces the need of additional storage capacity (B´odis et al. 2014). However, in Europe, sites suitable for large plants have already been exploited, and their environmental impacts are generally not well accepted. Mini/micro hydropower plants (net power input lower than 1 MW and 100 kW, respectively) are instead more sustainable and eco-friendly. They are becoming more attractive than big power plants, mainly in rural and decentralized areas for selfsustainment. In the industrialized countries, mini/micro hydropower is also useful for meeting the nonfossil fuel targets, whereas in developing countries, it may contribute to satisfy the rising demand of decentralized electricity. However, most low-head and low-discharge sites are currently not exploited because standard turbines cannot be employed economically in such conditions. As a consequence, there exists a 1

Ph.D. Candidate, Dept. of Environment, Land and Infrastructure Engineering, Politecnico di Torino, DIATI, Corso Duca degli Abruzzi 24, 10129 Torino, Italia (corresponding author). E-mail: emanuele.quaranta@ polito.it; [email protected] 2 Professor, Dept. of Environment, Land and Infrastructure Engineering, Politecnico di Torino, DIATI, Corso Duca degli Abruzzi 24, 10129 Torino, Italia. E-mail: [email protected] Note. This manuscript was submitted on December 5, 2015; approved on June 21, 2016; published online on August 16, 2016. Discussion period open until January 16, 2017; separate discussions must be submitted for individual papers. This paper is part of the Journal of Hydraulic Engineering, © ASCE, ISSN 0733-9429. © ASCE

demand for a cost-effective low-head hydropower converter, and water wheels may represent an attractive solution to this problem (Bozhinova et al. 2013; Müller and Kauppert 2004). This is especially supported by their fast payback periods and simplicity of construction. Water wheels fit well into the ecosystems; thus, they are considered not out of place when installed along a river. Water wheels can also contribute to the preservation of the cultural heritage (the restoration of old water mills), hence to the development of eco-tourism and promotion of social activities. Therefore, the employment of classical water wheels for the generation of renewable energy from water is becoming a cost-effective and sustainable solution in the mini/micro hydro field. Water wheels were introduced more than 2,000 years ago in order to produce energy, pump water, grind grain, forge iron, saw wood and stones, work with metal, and tan leather. Nowadays, the total number of existing historic small and micro hydropower sites in Europe is estimated to be around 350,000 (ESHA 2014). The oldest water wheel had a vertical axle, whereas the first kind, with a horizontal axle, was the stream water wheel (already described by Vitruvius in 27 BC). In stream water wheels, nowadays employed in flowing water and in sites with null or very low heads, the water flows below the wheel (Muller et al. 2007). Stream wheels are impulse machines; thus, the kinetic energy of water is mainly exploited, although recent studies have introduced considerable improvement, where also the hydrostatic force is employed (Gotoh 2001). In 1759, John Smeaton published experimental data (Capecchi 2013) demonstrating the higher efficiency of gravity wheels over the efficiency of impulse stream wheels. In gravity water wheels, the water acts by its weight, and the potential energy of water is mainly exploited. The stream kinetic energy can give its contribution during the filling time, when the water fills into the buckets and impacts against the blades. In the eighteenth and nineteenth centuries, some theories and experimental tests on water wheels were performed (Poncelet 1843; Morin and Morris 1843; Weisbach 1849; Bach 1886; Chaudy 1896; Garuffa 1897; Church 1914). However, theories were generally developed separately from experimental tests. Theories were often not validated, and several prescriptions on water wheels’ design were empirical and not based on scientific evidence. Furthermore, the experimental tests were

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carried out more than one century ago, with several uncertainties. At the end of the nineteenth century, the advent of modern turbines (such as Pelton, Francis, and Kaplan turbines, employed in big hydroelectric plants) replaced the classical water wheels, and energy production from low head sites was practically forgotten. Nowadays, because of the new interest in mini hydropower as described previously, the scientific research on water wheels is experiencing a revival. However, the engineering information on water wheels is ancient (hundreds of years), and there are still several uncertainties concerning their optimal working conditions, optimal geometric design, and fluid dynamic behavior. Hence, further work is necessary. Types of Gravity Water Wheels Among gravity wheels, overshot, breastshot, and undershot water wheels can be identified, depending on the water entry point with respect to the wheel. In gravity wheels, the potential energy of water is mainly exploited, whereas the flow kinetic energy effectively exploited by the wheel depends strictly on the hydraulic impact conditions and on the blades’ design (Quaranta and Revelli 2015b), such as blades number and shape. In overshot wheels (Quaranta and Revelli 2015a), the water fills the buckets along the downstream side of the wheel, entering from the top of the wheel. In breastshot water wheels (Quaranta and Revelli 2015b), the water enters into the wheel approximately in correspondence of the rotation axle. They can be divided in high and low breastshot wheels, when the water enters into the cells over the axle or below it, respectively. Breastshot wheels rotate in the reverse direction with respect to overshot wheels. The water is carried by the buckets to the outlet, downstream of the wheel, where it is released into the tailrace. Low breastshot wheels employed at very low head sites are called undershot water wheels (Quaranta and Müller 2016). In particular, breastshot water wheels are used in sites with head differences between 1.5 to 4 m, which are typical situations that can be found, for example, in irrigation canals. However, the most of the available design information is ancient (hundreds of years) (Poncelet 1843; Morin and Morris 1843; Weisbach 1849; Bach 1886; Chaudy 1896; Garuffa 1897; Church 1914), and very few fluid dynamics evidence is available in the literature, especially on the effects of the blades number on their performance. Scope of the Paper Based on the scientific gaps exposed in the previous sections, the main purpose of the present paper is the investigation of the effect of the blades number on the performance of a breastshot water wheel and to study, in more detail, the fluid dynamic interaction with the flow. The investigated wheel is 2.12 m in diameter, and it is a physical scaled model of a real one. The wheel will be better described in the “Method” section. Considering that an attractive opportunity for studying hydraulic problems and hydraulic machinery is now represented by computational fluid dynamics (CFD) simulations, CFD tools are used in this work to achieve the purpose mentioned previously. In the last decades, the employment of CFD tools for solving fluid dynamic problems has been developing more and more because CFD tools enable access to local flow properties with relatively low costs and with a substantial reduction in the experimental expenditure. For these motivations, fluid dynamic simulations may also represent a suitable and efficient method to investigate the hydraulic behavior of water wheels, obtaining useful information for their design. The involved phenomena are generally quite complex © ASCE

because they involve an unsteady 3D turbulent regime, a biphases formulation (a primary and a secondary phase, air and water, respectively), the gravity external force, and the moving and curved body of the wheel. CFD simulations for horizontal water wheels (Pujol et al. 2010) and stream wheels (Liu and Peymani 2015; Akinyemi and Liu 2015a, b) have been already presented in scientific journals, whereas to the best of the authors’ knowledge, CFD results for gravity water wheels can rarely be found. By the present work, the suitability of fluid dynamic simulations will be hence demonstrated also for breastshot water wheels; the numerical results are compared with those obtained by testing the same breastshot wheel (with 32 blades) in the laboratory, showing that the numerical solution can be considered accurate. The investigation of the optimal blades number for breastshot water wheels is justified by the fact that, in the scientific literature, the effect of the blades number on the performance of water wheels has been investigated mainly for stream water wheels. In Luther et al. (2013), the efficiency of a stream wheel with straight blades increased from 4 to 8 paddles, whereas in Tevata and Chainarong (2011), a water wheel with straight blades has been tested, and the efficiency decreased passing from 6 to 12 blades. The global behavior deducted from the last two papers suggests that for stream wheels, an optimum blades number can be identified. Moreover, in Müller et al. (2010), it is illustrated that the efficiency of a floating water wheel increased from 8 to 24 blades, with a substantial improvement from 8 to 12; the result also suggests that, in this case, an optimal blades number exists. Considering the previous deductions, the authors are, hence, stimulated to achieve similar results also for breastshot water wheels. In the following sections, the numerical setup and experimental method will be described. The numerical results will be compared with the experimental ones and discussed in detail. Practical engineering information concerning optimal numbers of blades for similar breastshot wheels in different hydraulic conditions will be reported.

Method A breastshot water wheel in an open channel was numerically investigated by CFD tools, in order to understand the effect of the blades number on its hydraulic behavior and performance, thus on the power output and efficiency. The diameter and the width of the tested wheel are D ¼ 2.12 m and b ¼ 0.65 m, respectively, and the height of the blades is 0.29 m; the investigated numbers of the curved blades are nb ¼ 16; 32; 48; 64. A physical model of the same dimension, with nb ¼ 32 and a total weight of W ≃ 3500 N, has also been installed in the Laboratory of Hydraulics at Politecnico di Torino, and the experimental results are used to validate the numerical ones. CFD Model: Geometry and Mesh The computational domain is constituted of three subdomains: the stationary domain of the channel, which conveys water to the wheel; the rotating domain of the wheel, which interacts with the channel; and stationary domain outside of the wheel, filled only with air. A sliding mesh approach was used. The stationary air domain is subdivided in an internal domain, in contact with the wheel, and an external domain. The latter has the scope to locate the boundary conditions sufficiently far away from the wheel and to stabilize the solution (Fig. 1). The channel and the wheel are meshed with tetrahedral elements, whose dimensions range between 0.01 and 0.02 m (a mesh

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Fig. 1. Computational domain and the boundary conditions (B.C.) of the numerical model; the figure represents a longitudinal section on the vertical symmetry plane; some representative dimensions are reported (m)

size of 0.02 m in the buckets ensures a mesh independent solution, as illustrated in the “Results” section). The stationary air domain is meshed with tetrahedron and cubic elements, whose dimensions range between 0.02 m near the wheel and the channel, up to 0.1 m at the boundaries of the external domain. The coarse mesh near the boundaries of the external domain does not affect the interaction between water and wheel. In order to reduce computational time without losing accuracy in the solution, it is not necessary to simulate the whole domain of the wheel (2π rad); half a wheel is sufficient to simulate the wheel-stream interaction (Barstad 2012), and the remaining half portion of the wheel is simulated with a coarser mesh and without blades (reducing the domain complexity). The final mesh has 1.3 million cells (Fig. 2).

The volume of fluid method (VOF) was used to solve the multiphase problem. This method can be used with two or more immiscible fluids. A continuity equation for phase q is solved, tracking the volume fraction αq of the phase throughout the domain [Eq. (1)] ∂αq ∂ðαq u¯ i Þ ∂ðαq u¯ j Þ ∂ðαq u¯ w Þ þ þ þ ¼0 ∂t ∂xi ∂xj ∂xw

ð1Þ

CFD Model: Simulation Setup The Reynolds Averaged Navier–Stokes (RANS) equations were employed for solving the flow field in the computational domains. The RANS equations include three momentum equations (one equation for each Cartesian coordinate) and the continuity equation. In the RANS equations, each variable y (pressure and the velocity) is decomposed in a time averaged value y¯ and a fluctuating component y 0 , the latter representing the difference between the instantaneous value of variable y and the time averaged value y¯ . Because the problem involves two phases (air and water, separated by a free surface), it is essential to add to the RANS equations an additional continuity equation to solve the multiphase problem. The additional continuity equation allows to determine in each cell of the domain the fraction volume of water and air. Knowing the volume fraction of each phase in all the cells of the domain, the physical properties (viscosity and density) of the mixture can be quantified [Eqs. (2) and (3)] and used to solve the RANS equations of the mixture. © ASCE

Fig. 2. Computational meshed domain; the figure represents a longitudinal section on the vertical symmetry plane; the portion of the wheel with the blades is meshed with finer elements, which after approximately 1 s, begin to interact with the stream

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where αq = volume fraction of phase q (the so called secondary phase, that, in this case, is represented by water) in each cell; αq ¼ 1 means that the cell is filled of water; αq ¼ 0 means that the cell is filled of air; 0 < αq < 1 means that the cell is near the free surface; xi , xj , and xw = directions of the Cartesian reference coordinate system; and generic u¯ y = time averaged velocity of the mixture [Eq. (5)] in the xy direction. Once the volume fraction of phase q is identified, the volume fraction of phase p (air) is calculated by αp ¼ 1 − αq. For the solution of the VOF equation, an implicit interpolation scheme was used, coupled with the level-set method, which is a well-established interface-tracking method for computing twophases flows with topologically complex interfaces. Because the level-set function is smooth and continuous (Osher and Sethian 1988), its spatial gradients can be accurately calculated. Therefore, the interface curvature and surface tension forces caused by the curvature are estimated accurately. However, the level-set method has a deficiency in preserving volume conservation (Olsson et al. 2007). On the other hand, the VOF method is naturally volumeconserved; it calculates the volume fraction of a particular phase in each cell, instead of the interface itself. The weakness of the VOF method is the calculation of its spatial derivatives because the volume fraction of a particular phase is discontinuous across the interface. To overcome the lack of the level-set method and the lack of the VOF method, a coupled level-set and VOF approach was used. After the determination of fraction volume αq and αp of the phases in each cell, the mixture properties are calculated: ρ ¼ αq ρq þ αp ρp

ð2Þ

μ ¼ αq μq þ αp μp

ð3Þ

where ρ and μ = density and dynamic viscosity of the mixture, respectively, and ρy and μy = properties of the generic phase y. The RANS continuity and momentum equations for the mixture are then solved. For an incompressible fluid the continuity equation is ∂ u¯ i ∂ u¯ j ∂ u¯ w þ þ ¼0 ∂xi ∂xj ∂xw

ð4Þ

The momentum equation for the mixture in direction xi , is 
∂ u¯ i ∂ u¯ ∂ u¯ ∂ u¯ ρ þ ui i þ uj i þ uw i ∂t ∂xi ∂xj ∂xw ¼ ρgi − þ

∂ p¯ ∂ ∂ þ μ∇2 u¯ i þ ð−ρu 0 i u 0 i Þ þ ð−ρu 0 i u 0 j Þ ∂xi ∂xi ∂xj

∂ ð−ρu 0 i u 0 w Þ ∂xw

ð5Þ

where ρ and μ = density and dynamic viscosity of the mixture; g = gravitational acceleration; p¯ = time averaged pressure; and u¯ i = mixture time averaged velocity along the direction xi . Analogous momentum equations are solved along the direction xj and xw . The terms ρu 0 i u 0 j are the Reynolds turbulent stresses, and they can be expressed as
 ∂ u¯ i ∂ u¯ j 2 0 0 τ i;j ¼ −ρu i u j ¼ μt þ ð6Þ − ρkδ ij ∂xj ∂xi 3 where μt = turbulent viscosity; k = turbulent kinetic energy; and δ ij = Kronecker delta. © ASCE

The turbulence viscosity is modeled using the shear-stress transport (SST) k − ω model (Menter 1994). In this model, the turbulent viscosity is expressed as a function of the turbulent kinetic energy k and the specific dissipation rate ω ¼ ϵ=k, where ϵ is the turbulence dissipation. Th...