Optimal design process of crossflow Banki turbines: Literature review and novel expeditious equations

1. Introduction

Hydropower is a renewable energy source, and it accounts for 16% of the global electricity generation, with an installed power capacity of 1330 GW in 2021 (International Energy Agency, 2021). Hydropower harnesses the energy of water to generate mechanical energy through the rotation of an hydraulic turbine. The mechanical energy is converted into electricity through an electric generator driven by the rotating turbine. Hydraulic turbines can be of different types: reaction turbines (e.g., Francis, Kaplan, Deriaz) mainly use the water pressure, impulse turbines (e.g., Pelton, Turgo, Banki) mainly use the water kinetic energy, and gravity turbines (e.g., water wheels and Archimedes screws) mainly use the weight of water (Okot, 2013; Quaranta and Revelli, 2018).

The Banki turbine, also known as Ossberger or Michel turbine, is the topic of the present study. It is composed of two main components, usually made of stainless steel (pers. comm. Italperfo s.r.l., 2021, pers. comm. Ossberger GmbH-Co, 2021, see details in Materials): a nozzle to control the flow entering the runner blades and the runner to extract the power from the flow (Adhikari, 2016). The runner is placed in a casing at atmospheric pressure, and it is connected to the generator through a shaft. A draft tube can be installed to exploit the residual head downstream and for in-conduit operation (e.g., aqueducts, Sinagra et al., 2020). The Banki turbine is a crossflow turbine because the water flow acts twice on the blades: the water jet interacts with the blades when they are near the nozzle (first stage), then the water jet flows from their outlet to the inlet of the same blades when they have reached the opposite side (second stage), as depicted in Fig. 1 (Adhikari, 2016).

The energy exchanged is typically 70–90% and 10–30% in the first and second stage, respectively. De Andrade et al. (2011) carried out computational fluid dynamic (CFD) simulations with the aim of highlighting the contribution of the two stages: they showed that 68.5% of energy is generated in the first stage, while in the second stage the remaining 31.5%. Adhikari and Wood (2018)demonstrated that the difference between the two stages is reduced as the flow rate increases: by doubling the flow rate, the percentages of energy produced in the first and second stages change from 88%-12% to 78%–22%, respectively. Woldermariam and Lemu (2019) found that the second stage contributes from 36.4% to 53.7%.

Full scale Banki turbines operate for a wide range of heads, from a few meters up to 200 m, and with low flow rates, from about 0.5 m3/s to 10 m3/s (Adhikari, 2016). The power capacity is between 10 kW and 1000 kW (Fig. 2). Recently developed designs, especially for enclosed pipes, can operate at heads >150 m and flows below 0.2 m3/s (Picone et al., 2021). The characteristic speed Ns, defined in Eq. (1), ranges from 60 to 200 (Restrepo, 2014):(1) $N_{s} = \frac{N P_{o u t}^{1 / 2}}{H_{n}^{1.25}} [-]$ where $N$ = runner rotational speed [rpm], $P_{o u t}$ = installed power [kW], $H_{n}$ = net head [m].

The maximum hydraulic efficiency is approximately 80% (Anand et al., 2021), and it is defined as the ratio of power output to power input:(2) $η = \frac{P_{o u t}}{P_{i n}} = \frac{M ω}{ρ g H_{n} Q} [-]$ where M = shaft torque [Nm], ω = runner angular velocity [rad/s], ρ = water density [kg/m3], g = acceleration of gravity [9.81 m/s2], Q = flow rate [m3/s]. The net head Hn can be approximately calculated as 0.94 H where H is the gross head (Nasir, 2013).

The main parameters that characterize the nozzle are depicted in Fig. 3(Sammartano et al., 2013): the nozzle width b, the nozzle throat (or depth) S0and the angle of attack α, namely the angle between the direction of water velocity and the tangent line to runner inlet. The nozzle profile has a particular shape to allow the flow to enter the runner always with the same angle α. The runner is characterized by the width B, the outer and inner diameters D1 and D2, the entry angle arc λ and the number of blades Nb, that are characterized by the radius ρb, the central angle δ and the inlet and outlet flow angle β1 and β2. β1and β2 are the angles between the tangent of the blade at the blade tip and the tangent to the external and internal circumference of the runner, respectively.

1.1. Flow rate regulation

The flow rate regulation system is depicted in Fig. 4 (Adhikari and Wood, 2018). It allows to open the turbine for one third, two thirds or three thirds, making the machine flexible towards the seasonality of the flow rates. A moving component further allows to change the flow passage area in the nozzle, for a fixed width opening (Adhikari and Wood, 2018), and it can be either a slider or a guide vane. The guide vane splits the flow in two, that undergoes a deceleration and a change in the flow trajectory, resulting in an irregular velocity profile to the runner inlet and a deviation from the optimal direction. This leads to an efficiency reduction. The slider, on the other hand, is a metal plate of semi-circular section that, sliding tangentially to the runner, acts on the amplitude of the entry arc angle. The water flow enters directly into the runner without undergoing a deceleration and without energy dissipation in the nozzle (Sinagra et al., 2014). Adhikari and Wood (2018) achieved the same maximum efficiency (88%) with the guide vane and the slider, but with the slider the turbine could maintain a high efficiency up to lower flow rates.

Mehr et al. (2019) and Sinagra et al. (2014) have highlighted the ability of the turbine to work well with flow rates as low as 16%–20% of the maximum flow rate, and this quality is linked to the regulation systems. The 16% can be reached by opening one third of the width of the turbine and then halving the flow rate by the nozzle. This has been confirmed in a real hydropower plant described in Cesoniene et al. (2020).

1.2. Why the Banki turbine?

In the recent years, new market, technological and social needs require hydropower to be more flexible, less environmental impactful and more cost effective, with new developments in existing infrastructures and in remote areas (Quaranta et al., 2020; International Energy Agency, 2021). Within this context, the Banki turbine represents an interesting technology and with new development opportunities, especially thanks to its simplicity in construction, flow rate/power regulation capacity and possibility to be installed both in traditional hydropower plants and in existing infrastructures, e.g. in aqueducts. In the former case, the plant has the typical configuration with inlet gate, penstock and turbine. In the second case, the Banki turbine is used as PRS (Pressure Reducing System): the turbine is installed in the pipeline with the dual function of dissipating pressures and generating energy. Additional applications of the Banki turbines are discussed in Appendix 4.

The manufacture of Banki turbines is quite simple compared to other types of turbines and can also be made in an artisanal way. The Banki turbines are basically made up of two circular rims joined together by welded blades. Therefore, the Banki turbine has great potential in rural areas, isolated and difficult to reach, in Non-Interconnected Zones and in low-income countries, but also in more industrialized countries (Ceballos et al., 2017). Some example are in Camerun (Ho-Yan e Lubitz, 2011), Tanzania (Mtalo et al., 2010), Pakistan (Chattha et al., 2014; Khan and Badshah, 2014), Bangladesh (Das et al., 2013), Nepal (Acharya et al., 2019), Myanmar (Win et al., 2016), Colombia (Durali, 1976). Fig. 5 depicts the distribution of the studies collected in the present manuscript. Das et al. (2013) compared a Banki turbine with a micro-Kaplan turbine of the same power and claimed that the Banki costs up to 7 times less in Bangladesh. 26 Banki turbines have been designed for 16 small hydropower plants in Bulgaria, with heads ranging from 22 m to 142 m and power up to 500 kW (Obretenov and Tsalov, 2021), while 6% of installed turbines in Saxony are Banki type (Spänhoff, 2014). In the European Union + UK (EU28), according to Voith Hydro database (pers. comm. of Markus Wirth), 24 Banki turbines are installed with a total power of 10.4 MW and maximum head 170 m.

Compared to the Francis turbine, the Banki turbine is able to guarantee an almost flat efficiency curve for a wider range of flow rates, although the maximum efficiency is generally lower (5–10 percentage points).

The life-span ranges from 40 to 50 years. Throughout its life, the Banki turbine does not require excessive maintenance, and this is also helped by the fact that the flow, passing through the runner, favors the self-cleaning capacity of the machine by continuously removing the sediments (pers. comm. of Ossberger GmbH-Co, 2021). In addition, as reported by Durali (1976) and in the technical specifications of the turbines produced by CINK Hydro-energy, the bearings are not in direct contact with the flow, but are protected by coatings, and can be easily lubricated and controlled, thus facilitating maintenance and increasing durability. In particular, the Ossberger company designs the bearings in order to guarantee a minimum operating life of 100 thousand hours, that is, more than 10 years (pers. comm. of Alberto Santolin, 2021). Finally, the study conducted by Adhikari et al. (2016) on a 7 kW turbine, showed that cavitation was found only in the second stage, minimizing cavitation erosion.

1.3. Design challenges and scope of the work

In Anand et al. (2021) the most recent review on the Banki turbine has been presented, with focus on its hydraulic behaviour and performance. The scientific challenges and gaps were also highlighted. A dataset collection of Banki turbines from literature was included, with the main geometric characteristics. Therefore, the reader interested in better understanding the Banki turbine behaviour and the literature research that has been carried out until year 2021 can refer to Anand et al. (2021). However, since a comprehensive design methodology for the Banki turbine has not been presented yet and it is not clear how to select some design parameters, the design gaps that are addressed in this study are the following.

1)
Rotational speed N: it can be generally estimated with an iterative process and choosing a certain value of D/B (diameter to width ratio of the runner). However, there is no systematic information on the optimal D/B value, and there are not expeditious tools able to easily estimate N for preliminary purposes. The only available equation to estimate the rotational speed is a function of the characteristic speed (Eq. (7)), but it does not effectively work over the entire range of operating conditions of the Banki turbine, since it is not dimensionless. This is also a lack encountered in similar equations for Francis and Kaplan turbines (Quaranta, 2019).
2)
Speed ratio SR = U/VU: this is defined as the ratio of U, the blade tangential speed, to Vu, the inflow water velocity existing the nozzle and projected along the U direction. The theoretical optimal value is SRth = 1/2, by applying the velocity triangle theory and finding the maximum power (Desai and Aziz, 1994; Das et al., 2013):

(3)

S R_{t h} = \frac{U}{V_{u}} = \frac{U}{V \cos α} = \frac{1}{2} [-]

with:(4)

V_{U} = c_{v} \sqrt{2 g H_{n}} \cdot \cos α [\frac{m}{s}]

(5)

U = ω \frac{D_{1}}{2} = \frac{2 π N}{60} \cdot \frac{D_{1}}{2} = \frac{π N D_{1}}{60} [\frac{m}{s}]

where α is the angle of attack, cv is the outflow coefficient, Hn is the net head [m], N is the rotational speed [rpm], ω is the rotational speed [rad/s] and D1 is the outer diameter [m]. cv is as a function of U/Vu and impeller inlet pressure (see Sammartano et al., 2016), and it can be assumed cv = 0.98 when the inlet is at atmospheric pressure.

Experimental studies show that the optimal SR is generally higher than the theoretical one SRth, but there are not practical suggestions on which SR value should be selected for the design.
3)
Number of blades Nb: different equations have been proposed in literature, but they exhibit some limitations: Khan and Badshah (2014) and Sammartano et al. (2016) are only valid in the investigated range, while Verhaart (1983) simply identifies the minimum number of blades to satisfy structural constraints and manufacture issues. Mockmore and Merryfield (1949) equation is an old empirical equation only valid within that study. These equations are better described in the Results section.
4)
Blade thickness: this parameter has been analysed in some papers (Verhaart, 1983; Khan and Badshah, 2014), but a complete methodology has not been proposed within the overall dimensioning process of a Banki turbine, and the related information has never been discussed in a systematic way. It must be noted that the blade thickness is not the only design parameter of blade design, as also the profile and the thickness variation along the blade are relevant (Sinagra et al., 2021).
5)
The shape ratio D1/b (outer runner diameter to the nozzle width) has never been adequately discussed, except in Khosrowpanah et al. (1988). In most of the studies, b = B (B=runner width), so that D1/b = D1/B. In other cases, B > b(see Appendix 3) to improve the efficiency, where b is estimated from the continuity of the discharged flow.

Additional issues not appropriately investigated in the literature are here addressed. Cost data were provided from hydropower companies and some Italian projects found on websites of local authorities. A section on the used materials is also provided thanks to information provided by companies, while the environmental performance (fish, sediment management, and cavitation problems) is described based on scientific literature data, writing a comprehensive section that aims at summarizing the fragmented literature information.

2. Review of design methodologies

In this section, the design methodologies available in literature are discussed. The first methodology was discussed in Mockmore and Merryfield (1949) and then adopted and improved by other authors (see next paragraph). Over the last decades, more advanced methodologies, e.g. those developed by Sammartano et al. (2013, 2016), optimize and complete the traditional methodology by proposing more complete equations, with optimized design parameters, especially for the blade angles and for the distributor profile. Methodologies based on iterative processes and on more design steps have been also introduced, that usually include Computational Fluid Dynamic (CFD) simulations (e.g., Hannachi et al., 2021; Mehr et al., 2019). However, these methodologies do not always solve the above mentioned gaps in an expeditious way, do not often provide generalizable results and are limited within the range of the investigated study.

Therefore, literature was surveyed, and relevant examples and data were compiled to assist the development of a new methodology, that was thus conceived to fill these gaps (most of the data are presented in Appendix 1 Investigated turbines to estimate the optimal rotational speed, Appendix 2 Investigated turbines to estimate the optimal tip speed ratio, Appendix 3 Investigated turbines to estimate the optimal shape ratio). The proposed methodology starts from the available equations or suggestions and improves the selection of the input parameters based on a critical review of literature data, expressing these input parameters as a function of other parameters when possible. In the following chapters some of the existing equations are improved, while new equations are introduced, replacing/complementing the old ones, e.g. those to estimate Nb, N and SR. These new equations were achieved considering only the studies where more values of these parameters were tested, so that the optimized ones could be selected and used in our analysis. Practical considerations on materials, costs, blade thickness tb and D1/b are also discussed.

2.1. Original methodology

Mockmore and Merryfield (1949) proposed the first design methodology, based on a jet impinging on one blade at time, and equipped with a guide vane for the flow regulation. This methodology was later adopted by other studies, e.g. Nasir (2013), Chattha et al. (2014), Achebe et al. (2020), Das et al. (2013), Acharya et al. (2019), Win et al. (2016), Mehr et al. (2019).

The maximum achievable efficiency η, from a theoretical point of view is (Mockmore and Merryfield, 1949):(6) $η = \frac{C^{2} \cdot (1 + Ψ) \cdot c o s^{2} α}{2} [-]$ where $C$ and $Ψ$ are loss coefficients that are generally set to 0.98 in the engineering practice (Mockmore and Merryfield, 1949). The angle of attack αtypically ranges from 16° (Michell, 1903; Mockmore and Merryfield, 1949; Durali, 1976; Khosrowpanah et al., 1988) to 24° (Fiuzat and Akerkar, 1989), and in some cases it is 22° (Desai and Aziz, 1994; Totapally and Aziz, 1994).

By assuming Ns = 513/Hn0.505 (Desai e Aziz, 1994; Penche, 1998; San and Nyi, 2018), Eq. (1) gives:(7) $N = \frac{513 \cdot H_{n}^{0.745}}{\sqrt{P_{o u t}}} [rpm]$

The diameter can be estimated from the speed ratio, i.e. Eqs. (3), (4), (5):(8) $D_{1} = \frac{82.9 \cdot S R_{t h} \cdot \cos α \cdot \sqrt{H_{n}}}{N} [m]$

The internal diameter D2 can be estimated as x $D_{1}$ , with x ranging from 0.65 to 0.68, in particular it was 0.65 in Sinagra et al. (2014) and Nasir (2013), 0.66 in Mockmore and Merryfield (1949) and Chattha et al. (2014), 0.67 in Adhikari (2016), 0.68 in Durali (1976), Khosrowpanah et al. (1988), Fiuzat and Akerkar (1989), Totapally and Aziz (1994) and Desai and Aziz (1994). In few cases it is 0.75 (Anand et al., 2021).

The inflow angle is:(9) $\tan β_{1} = 2 \tan α$

The outflow angle is instead generally set to β2 = 90° (Fig. 3), e.g., Mockmore and Merryfield (1949), Durali (1976), Fiuzat and Akerkar (1989), Khosrowpanah et al. (1988), Desai and Aziz (1994), Adhikari and Wood (2018) and Nasir (2013).

The depth of the water jet just upstream of the balde tip s was suggested to be:(10) $s = k \cdot D_{1} [m]$ with k = 0.075–0.1 (Mockmore and Merryfield, 1949), with a typical adopted value of k = 0.087 (Nasir, 2013; Mockmore and Merryfield, 1949).

Therefore, the distance between two blades t is:(11) $t = \frac{s}{\sin β_{1}} [m]$

Nasir (2013) suggested the following equation to estimate t, with α = 16°, β1 = 30° e k = 0.087:(12) $t = 0.174 D_{1} [m]$

The width of the nozzle can be calculated from the continuity equation:(13) $Q = V \cdot s \cdot b \overset{y i e l d s}{\to} b = 0.077 \frac{N \cdot Q}{H_{n} \cdot \cos α} [m]$

The curvature radius ρb of the blade profile is a function of the blade angles and inner and outer radius, R1 and R2, respectively:(14) $ρ_{b} = \frac{[{R_{1}}^{2} - {R_{2}}^{2}]}{2 (R_{1} \cos β_{1} + R_{2} \cos β_{2})} [m]$

The angle at the centre of the blade δ is:(15) $\tan \frac{δ}{2} = \frac{\cos β_{1} - \frac{R_{2}}{R_{1}} \cos β_{2}}{\sin β_{1} + \frac{R_{2}}{R_{1}} sen β_{2}} [-]$

The product $ρ_{b} δ$ gives the blade length Lc (m).

2.2. Recent methodologies

The modern design methods have improved the original design, especially the design of the blades and the nozzle arc, considering that more blades interact with the water jet along the entry arc angle $λ$ (Fig. 6, from Sammartano et al., 2013). CFD simulations are often carried out to further optimize the design parameters (Mehr et al., 2019). Most of these methodologies have been developed for applications in closed pipes, e.g. in aqueducts (Sinagra et al., 2020, 2021; Hannachi et al., 2021; Sammartano et al., 2017).

The angle of attack α is generally set at 22° (and the nozzle profile designed to maintain α along the nozzle arc), and the angle β1 is calculated as in Sammartano et al. (2013):