Volume 6 hydro power 6 16 – simplified generic axial flow microhydro turbines Volume 6 hydro power 6 16 – simplified generic axial flow microhydro turbines Volume 6 hydro power 6 16 – simplified generic axial flow microhydro turbines Volume 6 hydro power 6 16 – simplified generic axial flow microhydro turbines Volume 6 hydro power 6 16 – simplified generic axial flow microhydro turbines
6.16 Simplified Generic Axial-Flow Microhydro Turbines A Fuller and K Alexander, University of Canterbury, Christchurch, New Zealand © 2012 Elsevier Ltd All rights reserved 6.16.1 6.16.1.1 6.16.1.2 6.16.1.3 6.16.1.4 6.16.1.5 6.16.2 6.16.2.1 6.16.2.1.1 6.16.2.1.2 6.16.2.2 6.16.2.2.1 6.16.2.2.2 6.16.2.2.3 6.16.2.2.4 6.16.2.3 6.16.2.3.1 6.16.2.3.2 6.16.2.3.3 6.16.2.4 6.16.2.4.1 6.16.3 6.16.4 6.16.5 Further Reading References Introduction and Context What Is a Simplified Generic Axial-Flow Microhydro Turbine? Who Would Use Such a Turbine? Representative Designs Energy Alternatives and Unconventional Economics What Is Specific Speed? Component-Level Design Methods Water Supply Penstock Open flume Volute Runners need swirl Volute characterization Swirl predictability Limits of the tangential inlet volute Runner Specifying runner geometry An example runner How specific speed influences blade shape Draft Tube The effect of diffuser inlet swirl Turbine Selection from an Existing Range Direct Sizing Conclusions Nomenclature A area AR diffuser area ratio C penstock cost Cp pressure coefficient d diameter, penstock inner diameter FAM angular momentum flux factor g acceleration due to gravity h head loss H total head K1,2,3 penstock optimization constants Kd diffuser loss coefficient L penstock length LR diffuser length ratio _ mass flowrate m N runner speed (rev min−1), diffuser length NR Reynolds number NS specific speed p pressure P power Q turbine volumetric flowrate Comprehensive Renewable Energy, Volume 436 436 438 438 442 442 443 444 445 449 449 450 451 451 452 453 454 455 455 456 460 460 463 465 465 466 Qa available volumetric flowrate r radial position R diffuser wall radius T torque U blade velocity V velocity β local blade angle δ blade trailing edge deviation Δ change of indicated quantity η efficiency θ azimuth position in runner frame, diffuser half-angle ρ density ψ blade setup angle ω runner angular velocity Subscripts–Stations headwater between intake and penstock between penstock and volute doi:10.1016/B978-0-08-087872-0.00617-X 435 436 Design Concepts between volute and runner between runner and draft tube tailwater i component inlet o component outlet n normal r radial rel relative t tangential Subscripts–Pressures Subscripts–Turbine components d draft tube g generator i intake m transmission (mechanical) p penstock r runner t turbine composite T site total v volute Subscripts–Velocities a axial abs absolute d dynamic s static t total Subscripts–Runner blade location h hub LE leading edge t tip TE trailing edge Subscripts–Heads g gross n net 6.16.1 Introduction and Context 6.16.1.1 What Is a Simplified Generic Axial-Flow Microhydro Turbine? A simplified generic axial-flow microhydro turbine describes a class of devices for extracting energy from an elevated source of water, whose main design points are communicated in this chapter These devices are ‘simplified’ relative to large-scale turbines in terms of their geometry, construction, and operation They are designed to operate at peak efficiency at a single speed and flowrate with minimal maintenance Their geometry is fully fixed, the transmission is direct, and guide vanes are omitted The devices are ‘generic’ in that while each individual turbine is tailored to its site, the process for designing it is flexible enough to accommodate a range of inputs This is especially relevant for microscale turbines, which, being smaller, may be mounted entirely above ground on a foundation, reducing civil works complexity Best practices can be applied consistently, and consultation required for each turbine is reduced or eliminated ‘Axial flow’ describes the path the water takes through the turbine runner The runner is the device that extracts work by rotating under a torque From Euler’s turbomachinery equation, runner torque is equivalent to the change in angular momentum of the water For an incompressible fluid, this is proportional to the fluid’s change in rVt In an effort to reduce energy lost to residual swirl, most turbine designs use a stator to convert some of the static pressure to a tangential velocity, after which the runner reduces the swirl back to zero Turbine head is also proportional to this change in angular momentum, such that in general low-head turbines, which cope with less swirl, are of the axial-flow variety Axial flow corresponds to a high specific speed ‘Microhydro’ describes the size of a turbine in terms of its power output The product of head and flowrate is relatively small It does not mean microscopic, although relative to existing gigawatt-scale installations, microhydro turbines are roughly a million times smaller The term itself is a bit vague, and has been variously described as turbines producing from to 10 kW up to 35 to 100 kW In many cases, the term ‘microhydro’ could be replaced with ‘community scale’ or ‘household scale’ due to its frequent applications in those duties For this chapter, about 2–35 kW is the range of interest Starting from the descriptions above, the focus can be narrow further, beginning with the aim to design simple turbines Classically, one of the most complicated and expensive components of a small turbine has been the governor When hydraulic turbines are employed to generate alternating current (AC) electricity, the frequency of the generator’s electrical output is typically proportional to runner speed In this case, the runner speed must be strictly controlled to meet electrical supply quality standards under varying flow conditions Fortunately, the maturity of digital power electronics in recent decades has provided an economical answer to the challenge of governing small turbines: electronic ballast governors In a typical simple installation, the electronic controller senses the frequency output of the generator directly and if the frequency is above the target it switches in more resistance from a bank of Simplified Generic Axial-Flow Microhydro Turbines 437 resistors to slow it down If the frequency is too low, it reduces the generator’s electrical load and the runner speeds up This loop is performed several times per second so that the effects of discrete switching are not noticeable This balancing of real user loads with ballast loads ensures that the electrical quality is high at all times, ready for a user load to be switched on The requirement for the ballast resistors is to be able to absorb the turbine’s full output, although there is no reason the ballast load cannot contribute to a useful purpose such as water or space heating While the electrical design of a microhydro installation is of great importance to the operators, it is outside the scope of this chapter, which focuses on the hydraulic aspect of turbine design As turbine designers, we will simply assume that the generator is a directly driven four-pole induction motor and as such requires a nominal 1500 rev min−1 to produce standard 50 Hz AC output Most typically, microhydro turbines are considered to function as a run-of-river system, that is, they have no appreciable storage capacity Depending on the size of the parent stream, the turbine flow may be a significant portion of the total stream flow While an ideal, simplified stream would have a constant flowrate all year-round, real streams exhibit significant variation with the seasons and from year to year This fact is best illustrated by a flow duration curve (FDC), which is essentially a histogram of flowrate and shows what proportion of the total sample period the flowrate exceeded a particular value Figure is an example of a typical New Zealand mountain stream, where the flood flows are many times the dry season flow Fully fixed turbines in particular require a careful study of at least year’s hydrological data, and are generally given a design flowrate of the minimum expected stream flow less any required flows reserved for other purposes, which we will call the available flowrate, Qa, shown in Figure This does have the downside of not capturing available energy all of the time, but avoids the complications of automated governing and varying output, which is especially important in situations where the water turbine is the sole source of electricity for its users In the absence of adjustable turbine geometry, the turbine flowrate will be a unique function of turbine head As the speed is fixed by the governor, and the site head will not likely vary appreciably, the scale of the turbine needs to be such that the design flowrate is passed under these precise conditions, otherwise the relatively narrow peak of efficiency of the fully fixed turbine will never be utilized From the perspective of the turbine manufacturer, a fully fixed turbine greatly reduces mechanical complexity and simplifies construction However, the same decision works against the designer, who must ensure that the geometry specified will result in a turbine operating at peak efficiency Once built, there will be no way to recover from bad design In an effort to produce not a single efficient axial-flow turbine design, but a method for producing a uniformly efficient range of turbines, each step of design of each component must be left as flexible as possible For the ‘intake’ this means being able to cope with different sites’ requirements for a weir, penstock, or open flume without breaking compatibility with turbines unless absolutely necessary; ‘volute’ this means being scalable to different sizes and adaptable to different swirl requirements with high confidence in the resultant exit flow from a minimum number of template designs; ‘runner’ this means choosing leading edge blade angles by a method of zero torque from a valid upstream surface, choosing trailing edge angles for purely axial flow, and choosing a blade shape that lies on a plane, allowing the entire blade – traditionally one Camp stream: 2007 Flowrate, lps 1000 100 10 Qa Reserved flow 2007 0.2 0.4 0.6 Fraction of time flow exceeded Figure A typical mountain stream flow duration curve 0.8 438 Design Concepts of the most geometrically complicated turbine components – to be cut from flat sheet steel and welded directly to a cylindrical hub; and ‘draft tube’ this means avoiding the pitfalls of overly optimistic diffusers, understanding how the runner exit flow impacts the draft tube’s performance, and giving the draft tube appropriately weighted attention due to the amount of dynamic head it is responsible for recovering As are all geometrically related groups of turbine designs, axial-flow turbines are employed over a limited range of specific speeds where they perform more efficiently than competitive designs such as Pelton or Francis turbines Turbines that are exactly similar geometrically and dynamically, but differ only in scale, will have the exact same specific speed Specific speed will be defined later in detail, but it is helpful to think of it as a function of runner speed, turbine head, and turbine power which quantifies the general shape of the turbine with a single number To illustrate the two ends of the spectrum, turbines typified by ‘high head and small flow’, such as Pelton wheels, have a low specific speed, and ‘low head and large flow’, such as unshrouded free-stream turbines, examples of which are marine current turbines and wind turbines, have a high specific speed 6.16.1.2 Who Would Use Such a Turbine? Users of axial-flow microhydro turbines are a varied bunch, including • • • • communities looking for a first or more dependable or economical source of electricity; farmers, tourist lodges, camping huts, ski lodges, or other remotely located off-grid users with suitable streams; developed nations attempting to reduce environmental impact and utilize existing renewable resources; and wastewater treatment and irrigation works where low-head water is an inevitable by-product of the primary activity The sites may be relatively modest in absolute power terms, but are more than sufficient to be valuable to an individual or small group, where a typical Western lifestyle requires an average of about kW per person, although peaks are generally several times that Their low-head and run-of-river nature means a good site will not require extensive damming or flooding Next, let us take a look at some modern examples of this class of turbines 6.16.1.3 Representative Designs To illustrate the variety of designs under the heading of small, low-head hydro, some examples of recent commercial and academic work, including the authors’, are listed below Not all designs shown are strictly intended for microscale installations StraflowMatrix/HydroMatrix (546–700 kW, 5.5–30.5 m; VA Tech HYDRO GmbH) Square, self-contained turbine units designed for a fixed flowrate are stackable in rectangular arrays to meet various flowrate requirements The StraflowMatrix turbine unit is characterized by a unique generator configuration where the rotor coils are arranged around the periphery of the runner shroud, while the earlier HydroMatrix uses a more conventional bulb configuration [1] Flow is from left to right StraflowMatrix/HydroMatrix VA Tech HYDRO GmbH eKIDS (1–200 kW, 2–15 m; Toshiba) Japanese manufacturer Toshiba markets a four-model microhydro lineup generating from kW up The intended audience is mainly city utilities and waste management facilities where residual head exists in an industrial environment and the recovered Simplified Generic Axial-Flow Microhydro Turbines 439 energy can easily be injected back into the grid The limited English product literature states the turbines’ ability to operate off-grid [2] Flow is from right to left eKIDS Toshiba Siphon propeller turbine (10 kW, ∼3 m; IT Power) The UK firm IT Power has developed a belt-driven propeller turbine that operates fully above the headwater to reduce civil works complexity and protect the turbine from flooding damage, but requires priming by external means for start-up [3] Flow is from a headwater to the right, through the conical draft tube to the tailwater channel on the left Siphon propeller turbine IT Power VLH turbine (100–500 kW, 1.4–2.8 m; MJ2 Technologies S.A.R.L.) The Canadian-developed very low-head turbine is different from most other low-head designs Rather than attempting a smaller size and higher speed turbine to achieve a more compact machine, the designers have opted for a large through-flow area and low peak efficiency speed which keep runner exit velocities to a minimum, which avoids the need for a draft tube Water-to-wire efficiencies are about 79% for the smaller turbines Testing took place at the University of Laval [4] Flow is from upper right to lower left VLH turbine MJ2 Technologies Rotation axis of the structure Distributor and selfsupporting structure Direct drive variable speed PMG generator Turbine hub and kaplan runner 440 Design Concepts ‘Vaneless’ turbine (– kW, – m; Swiderski Engineering/Rapid-Eau) Turbine design has patent protection Designed with a low fish mortality rate in mind, the omission of flow-spanning vanes evidenced in this design should also translate to reduced susceptibility to clogging by entrained leaves and other trash, which is an important requirement for microscale turbines No test results are available Testing took place at the University of Laval Flow enters the volute through the rectangular opening and leaves for the runner to the left ‘Vaneless’ turbine Swiderski Engineering / Rapid-Eau Sub-kilowatt (0.3–1 kW, 1.5–4 m; Exmork/Energy Systems & Design/Yueniao) These four companies manufacture or supply a similar class of turbines: a one-piece turbine unit consisting of a runner and guide vanes connected to a directly driven generator by a rigid frame, which is meant to be directly fitted to a circular orifice in the bottom of an open race They are generally unregulated and produce either standard AC directly or DC power for charging battery banks Exmork (http://www.exmork.com) is a Chinese manufacturer whose products are distributed in Canada by PowerPal and in Europe by Kleinstwasserkraft Klopp PowerPal (http://www.powerpal.com) is a Canadian company that manufactures in Vietnam and imports Energy Systems & Design (http://www.microhydropower.com) is a Canadian company that produces a kW turbine similar to PowerPal’s Yueniao (http://www.yueniao.com/) is a Chinese manufacturer that produces a similar kW turbine Flow is top to bottom in the picture shown Sub-kilowatt Exmork/Energy Systems and Design/Yueniao PAT (– kW, – m; various) Using pumps as turbines (PAT, or BUTU in Spanish) provides an alternative to purpose-built turbines where pumps are available at competitive prices, and can cover essentially the same head and flow range as the pumps themselves High specific speed axial-flow pumps, which also operate at relatively high specific speeds as turbines, are more expensive per kilowatt than their lower specific speed centrifugal counterparts If the efficiency handicap of operating in reverse is overcome, the simplicity of a single unit containing turbine and generator is certainly attractive [5, 6] Flow enters the volute by the tangential pump outlet and leaves through the pump eye Simplified Generic Axial-Flow Microhydro Turbines 441 PAT various Archamedean screw (5.5–63 kW, 1.6–8 m; Western Renewable Energy) The UK firm Western Renewable Energy has developed the Archamedean screw as a relatively novel method of making electrical energy from a low-head source It is comparable to traditional waterwheels in that it requires considerable civil works and a significant speed increase in order to generate 50 Hz AC power, which came about when shaft output was directly used for mechanical work, which could utilize the slow rotational speeds directly Flow is top to bottom along the channel by parcels which are separated by the helical blade of the runner Archamedean screw Western Renewable Energy Giddens propeller turbines (1.4–4.1 kW, 3.4–10.7 m; University of Canterbury) The Mechanical Engineering Department of University of Canterbury has developed a range of radial-flow, mixed-flow, and, more recently, axial-flow propeller turbines called Giddens propeller turbines, named after late civil engineering professor Peter Giddens, who spearheaded the project, designed for forgiving construction and reliable use in remote locations As with all typical tangential inlet turbines, the flow enters tangentially and swirls around while gaining a radial component before exiting axially 442 Design Concepts Giddens propeller turbines University of Canterbury 6.16.1.4 Energy Alternatives and Unconventional Economics Users may be characterized by their access to alternative energy options, which may be superior to microhydro in terms of cost or available output The alternatives for those considering microhydro as a source of electricity are a mix of renewable and nonrenewable sources, and may be a subset of grid connection, diesel or gasoline generators, solar photovoltaic (PV) cells, and wind turbines The energy users may see • microhydro as one of several energy supply options, or • microhydro as the only energy supply option The distinction becomes particularly important in off-grid installations, where the retail cost of electricity is not available as a benchmark for project cost effectiveness While options such as grid connection or generators have a well-defined capital and ongoing cost, microhydro sites are less forgiving in some ways when it comes to developing a site to the needs of the users The total output is hard-limited by the site, not by available cash, and the capital and labor external cash cost of developing a microhydro site to the point that it is generating useful power will vary significantly depending on the resourcefulness of the developers and the local availability of materials, which cannot necessarily be stated for alternative sources This phenomenon is what is commonly referred to as unconventional economics 6.16.1.5 What Is Specific Speed? As mentioned earlier, when comparing different turbine designs, turbine-specific speed is a useful parameter for quantifying families of turbines, that is, turbines of similar shape but different size The rigorous nondimensional form of specific speed is shown in eqn [1]: pffiffiffi P nS ẳ p gH ị 5=4 ẵ1 However, as a parameter used extensively across the breadth of the topic of turbomachinery, it appears in various convenient forms The form adopted by the authors is shown in eqn [2]: NS ẳ p N Pr H5=4 ẵ2 where Pr denotes the shaft power out of the runner, before the transmission Specific speed has been shown to be the best dimensionless parameter for characterizing the general shape of a hydroturbine from only rotational speed, head, and flow, and therefore its suitability to a given flow regime From an efficiency standpoint, the efficiency of a turbomachine peaks at some finite specific speed and then decreases mono tonically from there as a consequence of handling larger flows, meaning a larger proportion of the total inlet head is dynamic head to which hydraulic losses are proportional The result is that the practical peak efficiency attainable for a given scale of output is lower at very high specific speeds [7] From a compactness standpoint, efficient high specific speed machines are physically larger per unit power and incur higher construction and transport costs in general than their higher head cousins This is a direct result of the site’s lower head, or power density, its unit of power per unit weight of water, from the formulation of potential energy for a parcel of water, Simplified Generic Axial-Flow Microhydro Turbines Hg ¼ d dt dEp g dV ẳ P g Q 443 ẵ3 where Ep is the potential energy and V is the volume 6.16.2 Component-Level Design Methods A turbine is simply a combination of individual components acting in concert; therefore, a design methodology can be presented for each individual component First, heavily used definitions are introduced Then, Sections 6.16.2.1–6.16.2.4 present the authors’ methods for designing or specifying each of the turbine’s main hydraulic components: intake and supply, volute, runner, and draft tube Although not treated here, the transmission and generator efficiencies can significantly reduce the final output power if not handled properly Base SI units are used unless noted The main components of a penstock-based scheme are introduced in Figure 2, along with the numbered stations which define the boundaries of each component for energy accounting purposes Several terms will be used more than others to refer to heads, and need defining here Site gross head is the vertical distance between the headwater and the tailwater Hg ≡ H0 − H5 ½4 Hn ≡ H2 − H5 ½5 Hr ≡ H3 − H4 ¼ ηi ηp ηv Hg ½6 net head is the total head across the turbine and runner head is the total head across the runner Unless mentioned, stated heads are total, that is, static plus dynamic A head loss is denoted by an h, with a subscript indicating either the particular component or the stations between which the loss is incurred For example, hp represents the head loss in the penstock, and h03 represents all total head losses upstream of the runner The concept of efficiency is integral to the development of a system such as a water turbine Its definition varies with the component or system it describes, but it is generally a ratio of work out to work in Components that have no work output, for example, the penstock, volute, and draft tube, not possess an efficiency in this strict sense but are given an efficiency symbol to show their effect on net head These components’ ‘efficiencies’ are defined as the total head loss they incur divided by the site gross head, except for the draft tube, whose exit dynamic head is also a loss ‘Intake’ efficiency is defined as minus the head loss incurred by the intake, including any grates or trash racks, divided by the site gross head, and is neglected in this treatment Headwater Intake Penstock Generator Transmission Volute Runner Tailwater Figure Penstock-supplied microhydro scheme components and stations Draft tube 444 Design Concepts ‘Penstock’ efficiency is defined as minus the head loss incurred in the penstock divided by the site gross head Minor losses are excluded ηp ≡ − hp Hg ½7 ‘Volute’ efficiency is defined as minus the head loss incurred in the volute divided by the site gross head ηv ≡ − Cpv pdi =ρg Hg ½8 ‘Runner’ efficiency is defined as the work output at the runner shaft divided by available hydraulic power It does not include the effect of bearing or seal losses ηr ≡ Pr ρgQHn ½9 ‘Draft tube’ efficiency is defined as minus the head loss incurred in the volute divided by the site gross head ηd ≡ − Kdpdi =ρg Hg ½10 ‘Transmission’ or mechanical efficiency is defined as the work done on the generator divided by the work received from the runner ηm ≡ Ps Pr ½11 ‘Generator’ efficiency is defined as the electrical power at the generator terminals divided by the mechanical power delivered by the transmission ηg ≡ Pe Ps ½12 Runner head can now be written as site head reduced by the plumbing component efficiencies Hr ¼ ηi ηp ηv ηd Hg ½13 Mechanical power extracted by the runner can be written as its hydraulic power input reduced by the runner efficiency Pr ẳ r gQHr ẵ14 And finally, the available electrical power can be written as the runner mechanical output reduced by the transmission and generator efficiencies Pe ¼ ηm ηg Pr ½15 Turbine efficiency is an oft-mentioned term that bundles the efficiency terms of the volute, runner, and draft tube t ẳ v r d ẵ16 A catch-all efficiency term may be defined as the product of all component efficiency terms, ηT ¼ ηi ηp ηv ηr d m g ẵ17 Pe ẳ T gQHg ẵ18 where the site’s electrical output is defined by 6.16.2.1 Water Supply If beginning upstream of the turbine, the first component encountered is the intake and supply The intake and supply are the hydraulic and structural interface between the turbine and its source of water Especially for low-head turbines, this construction component of a microhydro project has the potential to determine the project’s economic sensibility It is for this reason that the hydrology and site should be carefully examined, and the intake chosen to suit Simplified Generic Axial-Flow Microhydro Turbines 445 Let us begin by stating several important definitions and assumptions integral to the intake and supply design method presented here As the turbines of concern are fully fixed and will operate at single condition, the variability of a real site and hydrology need simplifying to this level of detail The site gross head, Hg, may be defined as simply the vertical distance between headwater and tailwater, since stream depth is typically small by comparison and will not vary much with flowrate Available volumetric flowrate, Qa, may be taken as constant due to the fixed-output design criterion, and is defined as the minimum expected stream flow minus any minimum-flow requirements, ecological or otherwise, imposed by regulation or common sense In this case, the minimum stream flow can be taken from the full-year hydrology record if the turbine will work year-round, or some subset of the full year if the turbine will operate only seasonally If the turbine will be relied upon only during the wet season, this will result in a higher design flowrate The consequence of this simplification is clearly that the turbine, even if operating efficiently, will not be able to ‘follow the peaks’ of flowrate, and hence power, throughout the seasons The upshot is that the users are able to rely on a constant output After a site is selected for penstock development, the penstock will have a known length, L The penstock material cost, C, is assumed to be proportional to material volume The head loss incurred in the penstock, h, is a variable to consider while optimizing penstock flowrate, Q, and penstock inner diameter, d The average slope of the penstock, S, is estimated from Hg/L For the purposes of optimizing the intake and supply, the turbine is idealized by a fixed-efficiency machine, which allows the product QH to represent the turbine power being optimized At the risk of oversimplifying matters, sites where axial-flow microhydro turbines are applicable will fall into one of two categories depending on the most convenient and economical method for transmitting the stream’s flow and head to the turbine: ‘lower head sites’ may make use of an open-flume arrangement where the turbine is mounted near headwater level and supplied directly by an open channel, exhausting to the tailwater through a draft tube which is entirely in suction, and ‘higher head sites’ may make use of a weir-and-penstock arrangement where a low dam forms a small reservoir upstream to ensure continuity of flow, while the backwater is allowed to escape through a pipe running down to the turbine, which sits near tailwater level, so that the draft tube is less critical These two categories are purposefully vague, in that intake selection is very site-specific, meaning that Hg and S give an incomplete description of the site, and idiosyncrasies of the site may become important Having said that, when designing the intake, the way in which the site head is developed is of fundamental importance The slope between intake and tailwater level is the key parameter besides head that determines the appropriate intake type When S < 0.25, the water supply cost as a fraction of the total project cost will probably be the main factor of the project’s return on investment For either type of supply, open flume or penstock, for a given head a gradually sloping section will be more expensive to develop than a steeply sloping one due to penstock length For example, if a useful head is developed over only a relatively flat length of streambed, the length of penstock, or alternatively the length of channel that must be extended along a contour, will be longer than if the entire head is developed at a compact site, the ideal case being a conveniently located waterfall This being said, open channels are generally more economical for conveying a large flowrate with a small total head loss for long stretches, while a penstock allows large pressure heads to be transmitted down steep slopes In microhydro installations that are not grid-connected, the retail cost of electricity is of little importance In this case, where energy is needed and not optional, projects may be further divided into two categories of relative stream size, depending on whether ‘available stream power is more than desired’ and the turbine may be sized to provide sufficient power, bypassing much of the total flow, that is, Q < Qa, or ‘available stream power is less than required’ and the turbine design must be carefully optimized to produce the most energy possible, that is, Q = Qa This dichotomy highlights whether the project developers are in the luxurious position of having surplus stream power at their disposal A site whose total power is on the verge of being insufficient may increase its turbine head slightly by further investment in larger penstocks, although this will be less cost effective compared to the case of a surplus site where the entire turbine could be scaled up slightly to accommodate more flow While a site offers a fixed potential, developers determine how efficiently and economically power is extracted primarily by considering their cost constraints Depending on the site, the intake and supply may be more or less important in the total project cost, but regardless of cost, the hydraulic design of either the supply penstock or open channel will be crucial for the hydraulic performance of the turbine downstream 6.16.2.1.1 Penstock Assuming that a weir and penstock supply makes sense for a given site (Hg, Qa, and S), d and hp/Hg are the key controlling parameters of penstock cost and available hydraulic power The following analysis is an adaptation of Alexander and Giddens’ work [8] on penstock optimization Importantly, it can be shown that for a given d, the product QH is maximized when hp/Hg is 1/3 From the definition of available turbine head shown in eqn [19] H ẳ Hg hp ẵ19 and penstock head loss due to wall friction shown in eqn [20] hp ẳ K1 LQ2 ẵ20 446 Design Concepts where K1 characterizes the pipe and is defined in eqn [21] K1 ẳ 2g f 2 ẵ21 d5 turbine power shown in eqn [22] P ¼ K2 QHn ½22 may be rewritten as a function of the penstock losses as shown in eqn [23] P ¼ K2 QHg K1 LQ2 ị ẵ23 where K2, defined in eqn [24], characterizes proportionality of turbine power to the product QHn and turbine efficiency, ηt, is assumed constant K2 ¼ ηt ρg ½24 Differentiating the polynomial in eqn [22] with respect to Q gives the second-order polynomial equation [25] dP ¼ K2 Hg − 3K1 K2 LQ2 dQ ½25 into which eqn [20] rearranged to give K1 may be substituted back, giving eqn [26] dP ¼ K2 ðHg − 3hp Þ dQ ½26 Setting this expression equal to zero reveals that, by assuming a turbine efficiency independent of flow conditions, maximum turbine power for a given diameter penstock simply occurs when Q is such that hp/Hg = 1/3 This is a somewhat counterintuitive message that it is optimum to lose a third of the available head considering that commercial hydro schemes are designed to have penstock losses on the order of 5% This result is useful to select the most cost-effective penstock It may not be the developer’s choice, but it simply shows that it will cost more to use more of the site’s head Of more general use is to consider the site fixed as before, but choose d in the interest of controlling cost or maximizing power A methodology will be presented that accomplishes both, but a key point to make is that if the available stream power is near the turbine power being developed, the method should handle this discontinuity realistically To begin with, and before the analysis gets too bogged down in mathematical details, it is helpful to develop a computer tool that allows the developer to calculate results over a range of diameters for varying sets of inputs As far as the authors are aware, there is no information in the literature concerning precise optimization of hydro scheme intake and supply, the likely reason being that as has been pointed out, even on a large scale, the natural character of sites and hydrology escape neat characterization by one or two parameters Even less regular are the characteristics of the developers’ material supply and possible economies achievable locally and their labor force Building on the power maximization developed over eqns [19]–[26], let us highlight the key points of penstock selection In an effort to maintain simplicity and transparency, only losses due to wall friction are considered and any turbulent development length near the inlet is assumed to be small, both of which result in optimistic estimates of power and efficiency The penstock should be kept free of any unnecessary bends or lossy fittings to avoid additional losses For estimation purposes, penstock cost is assumed to be proportional to material volume and wall thickness is assumed to be proportional to inner diameter, such that penstock material cost can be defined as in eqn [27] ẵ27 C ẳ K3 Ld2 where K3, which has the unit of cost per volume, is defined as in eqn [28] π K3 ¼ CV d 2 ! −1 with CV the cost per unit volume of penstock material and the penstock outer diameter The general method for each value of d is as follows: Let Hg and Qa define the site Define a target value for hp/Hg of 1/3 from the above analysis Calculate Q required to achieve the target penstock loss using Moody diagram or equivalent However, if Q > Qa, set Q = Qa and calculate resultant hp and update Hn Regardless, calculate turbine power from eqn [23] Calculate penstock cost from eqn [27] Repeat for a range of d ½28 Simplified Generic Axial-Flow Microhydro Turbines 447 Point refers to the discontinuity mentioned earlier, the essence being that if the penstock is made large enough that the full available stream flow Qa does not incur the target head loss, then Q is limited to Qa and the head loss will simply be lower than the optimum value The implications on cost of reducing the value of hp/Hg will be apparent in the results To illustrate the use of the results in decision making, the sample calculation from Reference 8, has been reworked using this method The important parameters are Hg = 12 m, Qa = 0.090 m3 s−1, L = 72 m, K3 = 1764 $ m−3, and surface roughness ε = 0.1 mm Furthermore, to reflect the fact that only a range of discrete diameters are available, the commonly available diameters given in Reference are represented by circles along a continuous line in each of the figures that follow Figure illustrates the point that when d is large enough that Q is limited to Qa while attempting to maintain optimum hp/Hg, at still larger diameters it will be less than the target value Let us call the diameter where Q = Qa and hp/Hg is the target value as the critical diameter, d′ In Figures 3–8, d′ = 0.184 m Figure shows the same Figure Head loss in penstock due to wall friction 12 Turbine head (m) 10 Hn 0.1 0.15 0.2 0.25 Penstock inner diameter (m) 0.3 ... Subscripts–Runner blade location h hub LE leading edge t tip TE trailing edge Subscripts–Heads g gross n net 6. 16. 1 Introduction and Context 6. 16. 1.1 What Is a Simplified Generic Axial- Flow Microhydro. .. [5, 6] Flow enters the volute by the tangential pump outlet and leaves through the pump eye Simplified Generic Axial- Flow Microhydro Turbines 441 PAT various Archamedean screw (5.5 63 kW, 1 .6 8... for a parcel of water, Simplified Generic Axial- Flow Microhydro Turbines Hg ¼ d dt dEp g ρ dV ¼ P g ρQ 443 ½3 where Ep is the potential energy and V is the volume 6. 16. 2 Component-Level Design