Limitations on Tidal-in-Stream Power Generation in a Strait by Joel Atwater B.A.Sc., The University of British Columbia, 2005 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Applied Science in The Faculty of Graduate Studies (Civil Engineering) The University of British Columbia (Vancouver) March, 2008 c Joel Atwater 2008 Abstract In the quest to reduce the release of carbon dioxide to limit the effects of global climate change, tidal-in-stream energy is being investigated as one of many possible sustainable means of generating electricity. In this scheme, turbines are placed in a tidal flow and kinetic energy is extracted. With the goal of producing maximum power, there is an ideal amount of resistance these turbines should provide; too little resistance will not a develop a sufficient pressure differential, while too much resistance will choke the flow. Tidal flow in a strait is driven by the difference in sea-level along the channel and is impeded by friction; the interplay between the driving and resistive forces determines the flow rate and thus the extractible power. The use of kinetic energy flux, previously employed as a metric for extractible power, is found to be unreliable as it does not account for the increased resistance the turbines provide in retarding the flow. The limits on extraction from a channel are dependant on the relationship between head loss and velocity. If head loss increases with the square of the velocity, a maximum of 38% of the total fluid power may be ex- ii Abstract tracted; this maximum decreases to 25% if head loss increases linearly with velocity. Using these values, the estimated power potential of BC’s Inside Passage is 477MW, 13% of previous assessments. If a flow has the ability to divert through a parallel channel around the installed turbines, there are further limits on production. The magnitude of this diversion is a function of the relative resistance of impeded and diversion channels. As power extraction increases, the flow will slow from its natural rate. This reduction in velocity precipitously decreases the power density the flow, requiring additional turbine area per unit of power. As such, the infrastructure costs per watt may rise five to eight times as additional turbines are installed. This places significant economic limitations on utilityscale tidal energy production. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1 Resource Assessment Publications . . . . . . . . . . . . . . . 9 2.2 Methodology Publications . . . . . . . . . . . . . . . . . . . . 12 . . . . . . . . . . . . . 27 3.1 Initial and Boundary Conditions . . . . . . . . . . . . . . . . 28 3.2 Response of Generation on Velocity . . . . . . . . . . . . . . 29 Linear Drag . . . . . . . . . . . . . . . . . . . . . . . . 30 Table of Contents List of Figures 1 Introduction 2 Literature Review 3 Extractible Power from a Single Channel 3.2.1 iv Table of Contents 3.2.2 Quadratic Drag . . . . . . . . . . . . . . . . . . . . . . 30 Power Extraction . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3.1 Linear Drag . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3.2 Quadratic Drag . . . . . . . . . . . . . . . . . . . . . . 34 3.4 Approximation of Sinusoidal Forcing . . . . . . . . . . . . . 35 3.5 Variation in Turbine Behaviour . . . . . . . . . . . . . . . . . 37 3.6 Characteristic Region - British Columbia’s Inside Passage . 38 3.7 Costs of Production . . . . . . . . . . . . . . . . . . . . . . . 44 . . . . . . . . . . . . . . 47 4.1 Head Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2 Characteristic Region - Current Passage, British Columbia . 51 4.2.1 Estimation of Bottom Friction . . . . . . . . . . . . . 52 4.2.2 Resource Inventory . . . . . . . . . . . . . . . . . . . 52 . . . . . . . . . . . . . . . . 56 5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.3 4 5 Extractible Power from a Split Channel Conclusions and Recommendations v List of Figures 1.1 Boundary conditions for tidal and river flow . . . . . . . . . 2.1 Example locations for tidal power generation from Tarbot- 6 ton and Larson (2006). (Generated from longitude/latitude positions) 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Channel flow properties with no artificial energy extraction. Recreated from Bryden et al. (2004) . . . . . . . . . . . . . . . 2.3 13 Influence of proportional extraction on the mean flow speed. Recreated with modification from Bryden et al. (2004) . . . . 2.4 11 14 Schematic of flow through turbine or turbines, denoted by T, occupying fraction of cross-sectional area of channel. Recreated from Garrett and Cummins (2004) . . . . . . . . . 16 2.5 Solution of Equation (2.9) for = {0.1, 0.5, 0.9} . . . . . . . . 18 2.6 Ratio P/Pmax as function of δ1 (linear resistance). Recreated from Garrett and Cummins (2004) . . . . . . . . . . . . . . . 21 vi List of Figures 2.7 The scaled maximum power as a function of a parameter λ0 , representing the frictional drag associated with the turbines, for various values of n where the turbine drag is assumed proportional to the nth power of the current speed. Recreated from Garrett and Cummins (2005) . . . . . . . . . 3.1 Arrangement of the scenario under investigation. The location of the turbines is shown in grey. . . . . . . . . . . . . . . 3.2 24 28 Reduction in velocity as a function of increased turbine resistance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Extraction efficiency (η) as a function of kt /kf . . . . . . . . . 34 3.4 Inside Passage with measurement locations. Image source: NASA Visible Earth . . . . . . . . . . . . . . . . . . . . . . . . 3.5 39 ∆h between Victoria, BC and Port Hardy, BC and u at Station J09 (50 ◦ 22.400000 N 125 ◦ 39.500000 W) as a function of time (subset of data) . . . . . . . . . . . . . . . . . . . . . . . 3.6 Assumed characteristic velocity distribution over depth at DFO station J09 . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 41 Correlation of head loss with volumetric flow rate in BC’s Inside Passage . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 40 42 Correlation of head loss with the square of volumetric flow rate in BC’s Inside Passage . . . . . . . . . . . . . . . . . . . . 43 vii List of Figures 3.9 Power as a function of time in BC’s Inside Passage (subset of data) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.10 Required increase in turbine swept area per unit of power production as a function of installed turbine capacity. . . . . 4.1 46 A tidal channel connected to two infinite oceans; an island separates the channel and tidal energy conversion devices are installed in one sub-channel. . . . . . . . . . . . . . . . . 4.2 48 Extraction efficiency for various values of α and β where γ = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.3 Current Passage in Johnstone Strait . . . . . . . . . . . . . . . 53 4.4 Extraction Efficiency vs. α for Current Passage, British Columbia, a split tidal channel. β ≈ 1.0, γ ≈ 2.6 . . . . . . . . . . . . . . 4.5 53 Changes in total and channel flow rates as a function of turbine resistance α. . . . . . . . . . . . . . . . . . . . . . . . . . 54 viii List of Symbols A f F g ∆h ht kf kt KE L p P Q Q0 Rh t u z ρ ω Area Bottom roughness coefficient Force Gravitational acceleration Water surface elevation difference across channel Head drop across turbine Natural friction coefficient Turbine resistance coefficient Kinetic energy Channel length Pressure Power Volumetric flow rate Natural volumetric flow rate Hydraulic radius Time Velocity Water surface elevation Fluid density Tidal forcing frequency ix Acknowledgements I’d like to acknowledge the invaluable contribution of Dr. Greg Lawrence who provided invaluable assistance throughout this project. Thank you to Rob Millar for editing, feedback and review. To Kor, my friends and family, thank you for your support and putting up with all the time I’ve spent at the computer over the last couple years. x Chapter 1 Introduction Humans have long used external sources of energy to help them survive; fire, domesticated animals and water wheels were the early sources which developed into a myriad of schemes from wind-mills to steam engines to nuclear reactors. In recent centuries, the largest source of energy has been the burning of various forms of decomposed plant matter: fossil fuels, primarily coal, oil and natural gas. Fossil fuels have provided an easily released, dispatchable, abundant, transportable, high-density and low-cost energy source. Rising prices, geopolitical challenges and concerns regarding global climate change however, have spurred interest away from fossil fuels and toward the development of renewable energy: energy that can be harnessed without depleting the finite resource (Sims et al., 2007). Various schemes exist for generating renewable energy for human consumption. These include solar, wind, hydro-electric, biomass, geothermal, wave and tidal generation (Sims et al., 2007). Unfortunately, none of these technologies has the potential to singularly replace fossil fuels as an energy source, either because of a limited resource or the intermittency of production; a combination of these technologies must be employed if society is to 1 Chapter 1. Introduction wean itself off oil. This thesis will focus on the use of tidal power. Tidal energy in various forms has been employed in some form for hundreds of years (Garrett and Cummins, 2004). In modern times, there are two general paradigms for extracting this energy: capturing the potential energy of the water due to changes in sea-level or capturing the kinetic energy of the flow. These two schemes will be referred to as tidal-barrage generation and tidal-in-stream generation, respectively. Tidal-barrage schemes typically involve building a dam across the entrance to a bay. A control structure allows seawater to move into the bay on the flood tide. At high tide, the control structure is closed, trapping the water in the bay as the outside water level drops. The head difference between the inside and outside of the bay then drives flow through turbines which in turn drives generators, producing electricity. There are a number of facilities in operation; the largest is in La Rance, France with an installed capacity of approximately 240MW (Garrett and Cummins, 2004). In Canada, a tidal-barrage plant operates in Annapolis, Nova Scotia with an installed capacity of 18MW (Hammons, 1993). Additionally smaller facilities have been constructed in Russia and China (Tidmarsh, 1983). There are several issues with tidal-barrage schemes that have likely made it politically impractical for new developments in North America. The primary concern is environmental. Dramatic changes in tidal flushing within a bay have massive ecological implications in often biologically sensitive estuarine regions; silt infiltration into the bay, detrimental effects 2 Chapter 1. Introduction to migratory marine life, increases in suspended solids, increased scour, difficulty with flood control and potential salination of on-shore groundwater are also of concern (Tidmarsh, 1983). Furthermore, the high capital costs associated with the construction of dams, control structures and turbines make tidal-barrages of somewhat limited economic viability. Despite the above, if the environmental movement shifts to focus solely on greenhouse gas reduction rather than habitat preservation, there may be a resurgence in interest in tidal-barrage generation. Tidal-in-stream generation is a newer concept that involves the capture of kinetic energy from a tidal flow. The advantages of this approach are that placing turbines within a flow is thought to have a more limited environmental impact due a smaller sea-bed footprint, and that the turbines are typically located away from estuarine waters (Gill, 2005). Furthermore, tidal-in-stream generation could potentially be employed in regions with a small tidal range where tidal-barrage generation would not be feasible. Tidal-in-stream generation is often considered to be analogous to wind and some hydro generation schemes. In some aspects, the comparison is valid while in others it is inappropriate; the extraction mechanisms are virtually identical while the physics of the flows are dramatically different. The extraction of kinetic energy from the moving sea-water is performed by a turbine placed in the flow. The majority of schemes utilize either a horizontal or vertical axis rotor. As the flow passes the turbine this rotor turns, driving a generator producing electricity. More novel schemes 3 Chapter 1. Introduction also exist to extract power, including utilizing an oscillating hydrofoil or biomimetic devices (Finnigan, 2006). Near the turbine, the flow behaves like it would near a wind turbine: fluid from upstream passes through turbine creating a differential pressure and a wake of reduced momentum that eventually mixes with the free-stream. As such, much of the technology and methodology in the development of turbines has been adapted directly from the wind and hydro industries However, the physics controlling the overall flow for tidal-in-stream power differ greatly from those of wind and micro-hydro, this has enormous consequences in the assessment of the available resource. Wind is driven by differential pressure due to various meteorological conditions, primarily heating and cooling of regions due to solar radiation. This causes a velocity field to develop between the Earth’s surface and the edge of the atmosphere. Wind turbines extract energy only from the very lowest region of the atmosphere which allows fast mixing with the higher velocity upper atmosphere; the wake, the region downstream of the turbine that has a momentum deficit, quickly dissipates. As such, the velocity field is reestablished and effects of a turbine are not measurable outside the near-field. Given this behaviour, a site can be selected on the basis of wind velocity alone, allowing production capacity to be calculated using the kinetic energy flux through the swept area of the proposed turbine. Tidal flows are also driven by a pressure differential: the difference in 4 Chapter 1. Introduction sea-level along a channel. In contrast to wind however, a tidal flow is constrained by a channel which limits the total amount of mixing of momentum. In essence, it may be possible to extract sufficiently large amounts of kinetic energy such that the overall flow will not be absorb the momentum deficit of the wake; this results in a reduction in the overall flow in the channel, limiting production. A second analogy is often drawn between tidal-in-stream generation and micro-hydro. Micro-hydro is installed within the confines of a river and extracts energy from the flow (either pressure head or velocity head depending on the type of turbine used). The boundary conditions of tidal and river flow, are very different. Tidal flows are driven by a head difference across the channel and the flow rate is primarily a function of frictional resistance. Flow in rivers has a fixed flow rate; friction determines the velocity and depth of the flow. As turbine resistance in micro-hydro is increased, the head difference along the reach increases until steady flow rate can be reestablished. The limits on this resistance, and thus the generation, are based on the limits of the allowable backwater effect. In a tidal flow, increasing turbine resistance results in a decrease in flow with the maximum head difference being the tidal range. This is illustrated in Figure 1.1. Given the limitation of applying methodologies taken from wind or hydro, there is not a reliable and industry accessible method to accurately inventory tidal-in-stream power resources. Resource assessment is requisite 5 Chapter 1. Introduction Without turbine resistance With turbine resistance (Flow Reduced) (a) Tidal Flow: turbine resistance reduces the flow rate but does not change the head difference Without turbine resistance With turbine resistance (Flow Unchanged) (b) River Flow: turbine resistance does not change the flow rate, but causes backwater effect. Figure 1.1: Boundary conditions for tidal and river flow for significant progress in the field as financing, energy policy, licencing and technology development are all functions of the limits of extraction. In the context of the motivation addressed above, the purpose of this thesis is to investigate the limits of power extraction. Chaptesr 2 will examine the present body of knowledge related the assessment of tidal-instram power resources. Chapters 3 and 4 will present extensible models allowing the determination of generation limits for a two commonly proposed generation scenarios: single and split channels. Chapter 5 will discuss the conclusions of this thesis and present a roadmap for future research. 6 Chapter 1. Introduction There are significant challenges ahead in the quest to transition society to sustainable, renewable energy; the most fundamental of which is the selection of generation schemes. Central to ascertaining a scheme’s technical and economic viability is the assessment of the available resource. This thesis will investigate the necessary assessment methodology in the context of tidal-in-stream power generation. 7 Chapter 2 Literature Review A small number of articles have been published on the topic of tidal-instream energy resources. This literature generally falls into one of two categories: a. inventories of extractible power in a region based on surveys of the flow characteristics; and b. arguments based on fluid mechanics to determine the effects of extraction on a natural flow. The latter generally advance understanding within the field, while the former often do not accurately reflect the limits of power extraction. The primary metric for available power used in the majority of tidal energy inventories is the kinetic energy (KE) flux. This metric is relatively easy to tabulate as it is based only on the channel cross-section and natural velocities. However, little confidence can be placed in the KE flux model, as it does not predict the changes to the system as a result of installing power extraction devices that will retard the flow. 8 Chapter 2. Literature Review 2.1 Resource Assessment Publications There have been a number of initial resource assessments of tidal energy, typically limited to a specific region and funded by some authority from that region. In Canada, the primary assessment is Tarbotton and Larson (2006), produced for the Canadian Hydraulics Centre. This study identifies a large number of energetic sites, calculates the average KE flux through a cross section of the channel and presents this value as the tidal energy potential, identifying 42GW in Canada. While the use of KE flux is a logical first step, it has been refuted by Garret and Cummins (2004) and Garret and Cummins (2005) as being unreliable in estimating extractible power. Tarbotton and Larson (2006) does acknowledge the that it is impossible to extract all 42GW, however the discussion seems to suggest that the extractable portion is considered kin to turbine efficiency. From Tarbotton and Larson (2006): “The estimated Canadian tidal current energy resource is an indication of the potential energy available, not the actual power that can be exploited. Environmental, technological, climate and economic factors will determine what proportion of the potential can be utilised.” By stating that extractible power is some proportion of kinetic energy flux, implicitly assumes that KE flux is in some way related to extractable power, an assumption that has been shown to be unfounded in many 9 Chapter 2. Literature Review cases. Furthermore, Tarbotton and Larson (2006) do not specifically mention the Betz limit, a theoretical maximum stating that, at most, 16 27 (59%) of the kinetic energy from a cross section of fluid may be extracted (Bossanyi, 2001). In the assessment of the tidal energy potential, Tarbotton and Larson (2006) sum multiple kinetic energy fluxes along a single channel (an example is shown in Figure 2.1). This methodology is not well founded: by stating that the entire KE flux is available to be extracted implicitly requires that the fluid velocity will go to zero downstream of the turbine in question. If the velocity downstream of the turbine is zero, there will be no kinetic energy available at the next cross section; the summation implies, however, that the entire natural flux downstream can be captured as well. Further assessments were completed by Hagerman et al. (2005) in the United States, Abonnel et al. (2005) in France and The Met Office et al. (2004) in Great Britain. These studies primarily focused on determining velocity fields within territorial waters without summation of the resource. As such, these studies made no attempt to identify the total resource, though Abonnel et al. (2005) and Hagerman et al. (2005) did address local limitations such as the Betz limit and velocity gradients are developed due to bottom friction. 10 Chapter 2. Literature Review Upper Rapids 1 Upper Rapids 2 Surge Narrows Figure 2.1: Example locations for tidal power generation from Tarbotton and Larson (2006). (Generated from longitude/latitude positions) 11 Chapter 2. Literature Review 2.2 Methodology Publications An analysis by Bryden et al. (2004) uses a momentum balance to predict the changes in flow with small energy extractions. The authors consider a slug of fluid moving through a channel. Assuming steady state, the pressure gradient produced by the tidal head is balanced by advection and a retarding force of natural friction and the extraction devices such that: pressure gradient advection { z }| { bottom friction power extraction z }| { z}|{ ∂h ∂ 2 = − U (AU ρ) − U Per τ0 − ρgU A Px A ∂x ∂x z }| (2.1) where U is the fluid velocity, A is the cross sectional area, Per is the wetted perimeter, τ0 is the natural shear stress and Px is the power extraction per unit volume (W/m3 ). Applying conservation of mass for steady flow with no lateral inflow, such that ∂Q ∂x = 0 Equation (2.1) can be expressed as: Q2 1− 3 2 hbg ∂h ∂b Q2 1 Px A = − Per τ0 − 2 3 ∂x ∂x gh b ρgbh ρgQ (2.2) Bryden et al. (2004) then integrate Equation (2.2) with Px = 0 to determine the surface profile for an arbitrary Q. Various solutions of (2.2) are iterated to determine the flow that corresponds to the tidal head under investigation. The power extraction term is then increased and the relationship solved as before (Figure 2.2). The author ends the analysis 12 Chapter 2. Literature Review at this point; the primary goal of the paper was showing that the flow does indeed slow when additional retarding force is added (Figure 2.3). The relevant figure in Bryden et al. (2004) does not show that the flow quickly deviates from its initial linear trend at high levels of extraction. Using an arbitrary channel geometry, the authors calculate that extracting 10% of the KE flux will reduce the flow by 3% and extracting 20% will reduce the flow by 6%. It is then suggested that extraction of 10% of the KE flux could be considered a ‘rule of thumb’ for future development of tidal power. Bryden et al. (2004) do not provide any rationale for the value of 10% extraction. 40.1 2.50 40.0 Depth Velocity Depth (m) Location of Conversion Devices 39.7 39.6 2.00 1.50 39.5 1.00 39.4 39.3 Velocity (m/s) 39.9 39.8 0.50 39.2 39.1 0 500 1000 1500 2000 2500 3000 3500 0.00 4000 Distance (m) Figure 2.2: Channel flow properties with no artificial energy extraction. Recreated from Bryden et al. (2004) The formulation of the momentum equation used in this paper does not lend itself to the estimation of total extractible power as Equation (2.2) relies on a Px to represent the turbines. The magnitude of power extraction is a function of the flow and the tur13 Chapter 2. Literature Review Change in flow speed Influence of Energy Extraction on Current Speed 0% 0% -5% -10% -15% -20% -25% -30% -35% -40% -45% -50% 5% 10% 15% 20% Shown in publication 25% 30% 35% 40% Not Shown in publication Figure 2.3: Influence of proportional extraction on the mean flow speed. Recreated with modification from Bryden et al. (2004) bine characteristics. In formulation used by Bryden et al. (2004) however, the power extraction term is a control for the flow. In order to determine the maximum extraction, (a topic not addressed by the authors) one must slowly increment Px until the model fails. Furthermore, since the controlling parameter is dimensional, it is difficult to apply the results to multiple scenarios. In essence, this parameter is not particularly useful in the determination of maximum power extraction in the general case. However it can be used to determine the effects of small, fixed amounts of extraction. Despite the limitations of using the Bryden et al. (2004) approach for the estimation of total extractable power, a very important aspect is addressed in that the flow speed will be reduced with the installation of extraction devices. Furthermore, the authors remind readers that the design of extraction devices is largely dependent on the flow velocities for the determination of optimum geometry. Tidal energy developers must take into account the reduced velocities and not base their designs on the 14 Chapter 2. Literature Review natural conditions. More rigorous analysis that addresses total extractible power was conducted by Garrett and Cummins (2004) and Garrett and Cummins (2005). Garrett and Cummins (2004) provides some key insights as to the behaviour of tidal flows into and out of bays while Garret and Cummins (2005) addresses flows in straits. The first point presented by Garrett and Cummins (2004) is to remind the reader that for a turbine to extract power it must experience a pressure difference and must have flow passing through it; the absence of either of these conditions will result in zero power production. In order to perform work, a fluid requires both flow and a differential pressure as shown by: dW = W =F ·d→ dt Z p ~u · n̂dA (2.3) where W is work, F is force, d is distance, t is time, u is fluid velocity, p is pressure acting upon the surface under consideration and A and n̂ the area and normal vector respectively of that surface. Garrett and Cummins (2004) argue that maximum power is produced when the turbines span the entire cross section of the tidal channel. A general situation is presented where a turbine is installed in a flow such that it covers some fraction of total cross sectional area, denoted by (Figure 2.4). Assuming steady flow and a constant cross section, continuity then 15 Chapter 2. Literature Review u2 u0 p1 T p0 u0 u1 p2 p3 Figure 2.4: Schematic of flow through turbine or turbines, denoted by T, occupying fraction of cross-sectional area of channel. Recreated from Garrett and Cummins (2004) requires u1 + (1 − )u2 = u0 (2.4) and thus u2 = u0 − u1 1− (2.5) where the subscript 0 denotes the region upstream of the turbine, 1 denotes the region through the turbine, 2 is the region to either side of the turbine and 3 is the region downstream of the turbine. Garrett and Cummins (2004) assume that there is not a significant transverse pressure gradient downstream of the turbine. This assumption is likely reasonable in the case of a turbine with a low blockage factor. Using Bernoulli’s theorem, the pressure drop accross the turbines is then given by: 1 p0 − p2 = ρ(u22 − u20 ) 2 (2.6) 16 Chapter 2. Literature Review and thus power given by 1 P = ρu1 (u22 − u20 ) 2 (2.7) Using Equation (2.7), power is compared to the situation where the turbines occupy the entire channel, such that: P0 = (p0 − p3 )u0 (2.8) P x(1 + x − 2x) = P0 2x + (1 − 3x) (2.9) This yields: where x = u1 /u0 . Given that if the turbine is to act as a turbine and not a pump, x must be be less than unity. Solving Equation (2.9) for 0 < x < 1 (Figure 2.5) shows that P/P0 ≤ 1 for all values of . This implies that power from a set of turbines occupying a fraction of the channel cross section will never be larger than from a set of turbines occupying the entire channel cross section. As such, it can be concluded that in determining the maximum power output, only the situation in which the entire channel area is impeded need be considered. Garrett and Cummins (2004) then focus on extractible power through a channel leading to a closed bay. In this scenario the difference in water level between the inside of the bay and the ocean drives a flow. This flow is resisted by a turbine force per unit mass, F , creating a dynamical balance 17 Chapter 2. Literature Review 1 0.9 0.8 0.7 P/P0 0.6 !=0.1 0.5 !=0.5 !=0.9 0.4 0.3 0.2 0.1 0 0 0.2 0.4 x 0.6 0.8 1 Figure 2.5: Solution of Equation (2.9) for = {0.1, 0.5, 0.9} . Recreated from Garrett and Cummins (2004). 18 Chapter 2. Literature Review given by: g(ζ − a cos ωt) = −F L (2.10) where g is gravitational acceleration, a and ω are the amplitude and frequency of tidal forcing respectively, t is time, L is the length of the channel and ζ is the water surface elevation inside the bay. Two models for friction are then applied, one linear with velocity and one quadratic with velocity such that F = r1 u or F = r2 u|u|. Continuity is applied to the channel/bay as: A dζ = Eu dt (2.11) where E is the cross sectional area of the channel and u is the velocity in the channel. Equations (2.10) and (2.11) are non-dimensionalised and combined with ζ = aζ‘, t = t‘/ω and u = (Aωa/E)u‘ giving: dζ‘ cos t‘ − ζ‘ = dt‘ δ1 (2.12) for the linear case where the non-dimensional resistance is: δ1 = r1 LAω gE (2.13) and for the quadratic case: dζ‘ | cos t‘ − ζ‘|1/2 sgn(cos t‘ − ζ 0 ) = dt‘ δ2 (2.14) 19 Chapter 2. Literature Review where the non-dimensional resistance is:: r2 LA2 ω 2 a gE 2 δ2 = (2.15) Solving the dynamical balance given in Equation (2.10) allows the calculation of power production using P = ρELF u giving P = 2δ1 1 + δ12 Pmax (2.16) Pmax (2.17) for linear resistance and P = 4δ2 dζ‘ dt‘ for quadratic. As can be seen in Figure 2.6, power production rises to a maximum as turbine resistance is added. Past this peak, production falls with added turbine resistance as the effect of the flow slowing becomes the dominant factor. An interesting additional conclusion is that the tidal range inside the bay is not necessarily significantly affected even at the maximum extraction rate; the head across the turbines is largely driven by the phase difference of the water levels inside and outside the bay rather than solely by the differences in sinusoidal amplitudes. As such, flushing of the bay can be maintained. 20 Chapter 2. Literature Review 1 0.9 0.8 0.7 P/Pmax 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 !1 Figure 2.6: Ratio P/Pmax as function of δ1 (linear resistance). Recreated from Garrett and Cummins (2004) A specific note is made by Garrett and Cummins (2004) as to the use of kinetic energy flux as a metric for available power, specifically stating: “Perhaps the safest conclusion is just that 1/2ρu3 is unreliable as a power estimate in a situation involving flow into a closed basin” The conclusions of this article have some significant implications with regards to resource assessment of tidal energy: • The generation of power from a turbine requires both flow and differential pressure. • Analysis may be simplified to as maximum generation occurs when all of a flow passes through the turbines. • For a closed bay, generation does not necessarily significantly reduce 21 Chapter 2. Literature Review tidal flushing due to a phase difference that develops between the inner and outer water levels. It is important to note that this last conclusion can not be applied to tidal straits (the focus of this thesis) as the head difference that drives the flow is dependent on ocean water levels and is not generally changed by the flow through the channel. Garrett and Cummins (2005) address the issue of generation in tidal straits using a similar approach. The momentum equation for this case is given by ∂u ∂ζ ∂u +u +g = −F ∂t ∂x ∂x (2.18) where the pressure gradient along the channel is balanced by some element of resistance, advection and acceleration. Garrett and Cummins (2005) then subjects this relationship to a variety of conditions. The initial test is to neglect natural friction and to balance the pressure gradient only with acceleration and turbine resistance (the authors acknowledge that this situation is unlikely to occur in reality.) As in Garrett and Cummins (2004) power generation is given by ρuAF , now integrated over the length of the channel and assuming a linear turbine drag giving Z P = L Z ρF Qdx = ρQ 0 L F dx (2.19) 0 22 Chapter 2. Literature Review yielding c dQ − ga cos ωt = −λQ dt (2.20) where λ is a term associated with the number of the turbines in the flow. Assuming that the flow is uniform across each cross-section, Equation (2.20) can be solved, giving the maximum extractible power as 1 1 1 P0 = ρg 2 a2 (cω)−1 = ρcωQ20 = ρgaQ0 4 4 4 (2.21) where a is the amplitude of the tidal head across the channel and Q0 being the natural flow rate. While a unique solution may be reached, the fundamental assumption that bottom friction is negligible is unrealistic as natural friction is dominant in many situations. Garrett and Cummins (2005) continue with analysis neglecting bottom friction but investigating non-linear turbine resistance with flow. It was found through numerical solutions of the momentum equation that maximum extractible power occurs with linear friction. Various power extraction values are presented. See Figure 2.7. It should be noted that resistance to flow is typically considered to be quadratic with velocity (n = 2) (Finnemore and Franzini, 2002a) Assuming the drag is quadratic with flow, the authors add the effects of bottom friction and exit losses from the channel. The friction term, λ, is separated into a turbine resistance and a friction resistance; the ratio of power generated by the turbines to that dissipated by friction is equal to 23 Chapter 2. Literature Review 1.0 n=3 n=2 0.8 n=1.5 P/P0 0.6 n=1 0.4 0.2 n=0.5 n=0.25 0 0 1 2 3 4 5 λ’ Figure 2.7: The scaled maximum power as a function of a parameter λ0 , representing the frictional drag associated with the turbines, for various values of n where the turbine drag is assumed proportional to the nth power of the current speed. Recreated from Garrett and Cummins (2005) 24 Chapter 2. Literature Review the ratio between the separated turbine friction parameter and the total friction parameter used in Equation (2.20). Considering situations where the natural friction coefficient is large, (thus friction is dominant, not acceleration), the ratio of extractible power vs friction-free extractible power approaches 0.86λ−0.5 . Using this result, the extractible power from a strait can be given by 0.38ρgQ1 ζ. Where Q1 is the natural flow rate and ζ is the head difference across the channel. This corresponds to a reduction in flow rate to 58% of natural. Garrett and Cummins (2005) make a thorough argument as to the maximum of extractible power using a fundamental momentum balance. While the approach is complete, it has two weaknesses: • it is somewhat difficult to understand the procedure. • the model is not easily adaptable for variable turbine behaviour. Present assessments of tidal energy have been quite limited and have been intended to develop a rough estimate of the resource. As such, some studies have lacked the scope to investigate the complexities of tidal extraction and have elected to use the kinetic energy flux method due to its ease of application, a topic specifically addressed by Tarbotton and Larson (2006). The highly mathematical approach used by Garrett and Cummins likely further inhibits adaptation of more complex assessment models in this context due to the time required to apply the methodology. In general, Garret and Cummins’ arguments are generally complete for 25 Chapter 2. Literature Review the cases they addressed, however they may be limited in application due to its highly mathematical nature and lack of easy adaptability. 26 Chapter 3 Extractible Power from a Single Channel In essence, tidal-in-stream power that is extracted from a single channel falls into one of two categories: • flow into and out of a bay • flow through a tidal strait While the mechanism of power extraction from these locations would likely be identical, the applicable boundary conditions and thus the methodology for resource assessment are significantly different. Garrett and Cummins (2004) provides an excellent analysis of tidal energy potential of a channel connected to a bay. As such, the discussion in this thesis will be limited to a tidal strait that is connected to two oceans, or two parts of the same ocean. 27 Chapter 3. Extractible Power from a Single Channel 3.1 Initial and Boundary Conditions Consider a long, narrow body of water that connects two infinite “oceans”. The water surface in these oceans is considered to be independent of the channel. The water level difference across the channel, denoted by ∆h = z0 − z2 , varies slowly such that at any instant the system may be considered steady state; this assumption holds if the period of the tide outside the channel is significantly longer than period of a shallow water wave moving through the channel. If velocities in the oceans are considered to be negligible; the flow through the channel is driven only by ∆h and retarded by friction and acceleration. It is assumed that this process is dominated by friction and acceleration may be neglected. The effects of rotation are also not being considered. The scenario is illustrated in Figure 3.1. u1 Ocean z0 Ocean Location of Turbines z2 Figure 3.1: Arrangement of the scenario under investigation. The location of the turbines is shown in grey. Given the above conditions, Bernoulli’s relation with head loss may be 28 Chapter 3. Extractible Power from a Single Channel applied such that: 0 0 u20 u21 0 0 u22 p2 p0 + + + z2 + + z0 − hf (u) − ht (u) = 2g ρg 2g ρg 2g (3.1) where p is pressure, u is velocity, z is fluid surface level, g is gravitational acceleration, ρ is fluid density, hf is the head loss attributed to natural friction and ht is the head loss attributed to installed turbines. The addition of the u21 on the RHS of (3.1) implies that the flow will separate at the exit and dissipate as a jet. 3.2 Response of Generation on Velocity The primary response of the system is dependent on the relationship between head loss due to friction and velocity. This relationship is typically considered to be quadratic with velocity (Finnemore and Franzini (2002a)) however some observations of natural flows show that this relationship can be also linear (see Mattsson (1995) and Section 3.6) Following Garrett and Cummins (2005), both linear and quadratic resistance formulations will be considered. Linear Drag Quadratic Drag hf (u) = kf u1 hf (u) = kf u21 (3.2) ht (u) = kt u1 ht (u) = kt u21 (3.3) 29 Chapter 3. Extractible Power from a Single Channel Additional relationships for turbine drag will be discussed in Section 3.5. 3.2.1 Linear Drag If linear drag is assumed, Equation (3.1) can be rearranged as: z0 − z1 = (kf + kt )u1 + At this point, the velocity head term, u21 2g u21 2g (3.4) will be neglected. This term accounts for the flow exiting as a jet into an ocean and typically occurs where the cross section of the flow is large, therefore the u2 term should be small. This yields u1 as a function of the tidal range, natural friction and the turbine coefficient. u1 can be expressed as: u1 = 3.2.2 ∆h kf + kt (3.5) Quadratic Drag If the bottom friction drag is quadratic with velocity, the u21 /2g term may simply be combined with kf . Equation (3.1) then becomes: z0 − z2 = (kf + kt )u21 (3.6) 30 Chapter 3. Extractible Power from a Single Channel Solving u1 as a function of ∆h, kt and kf gives: s u1 = ∆h kf + kt (3.7) The response in velocity to changes in resistance are shown in Figure 3.2; this change in flow has significant implications on power extraction. 1 Quadratic Drag Linear Drag u/umax 0.8 0.6 0.4 0.2 0 0 5 10 15 kt/kf 20 25 30 Figure 3.2: Reduction in velocity as a function of increased turbine resistance. 3.3 Power Extraction To determine the amount of power that can be extracted from a flow, consider the rate of work the fluid does on a turbine as: P = ρght Qt (3.8) 31 Chapter 3. Extractible Power from a Single Channel where ρ is the fluid density, g is gravitation acceleration, ht is the head difference developed by the turbine and Qt is the flow passing through the turbine and given as: Qt = ut At (3.9) where ut is the velocity of the fluid passing through the turbine and At is the swept area of the turbine (Finnemore and Franzini, 2002b). Garrett and Cummins (2004) determined that maximum power can be extracted if the turbines occupy the entire cross section of the channel. As such, ut and At may be replaced with u1 and A1 respectively. This allows for the formulation of fluid power from a tidal flow as: P = ρgA1 u1 ht (3.10) It is now useful to define the term η, the extraction efficiency, as the fraction of the natural fluid power of the channel that may be extracted and given by: η= P ρgQ0 ∆h (3.11) where Q0 is the volumetric flow rate through the channel before any turbines are installed and ∆h is the tidal range given by z0 − z2 . 32 Chapter 3. Extractible Power from a Single Channel 3.3.1 Linear Drag Again, the analysis will be separated to consider linear and quadratic drag. Equations (3.5) and (3.10) are combined to give total power as: P = ρgA1 kt ∆h kf + kt 2 (3.12) The extraction efficiency, η can then be solved as: η= (kt · kf )2 (kf + kt )2 (3.13) Figure 3.3 plots η vs kt . To determine the conditions of maximum extraction, Equation (3.13) is differentiated such that: −(kt − kf ) · kf dη = dkt (kt + kf )3 (3.14) Setting Equation (3.14) to zero and solving for kt reveals that η reaches a maximum as kt = kf . This result can then be substituted into Equation (3.13) giving a maximum extraction efficiency of 14 . As such, the potential power of a strait can simply be expressed as: P = 0.25ρg∆hQ (3.15) Given this result, the effect on the flow at maximum production can be 33 Chapter 3. Extractible Power from a Single Channel 0.4 Quadratic Drag Linear Drag η 0.3 0.2 0.1 0 0 5 10 15 kt/kf 20 25 30 Figure 3.3: Extraction efficiency (η) as a function of kt /kf computed using: Qreduced kf = Q0 kt + kf As such, the flow will reduce to 1 2 (3.16) of its natural rate at maximum power extraction. 3.3.2 Quadratic Drag Assuming quadratic drag, combining Equations (3.7) and (3.10) gives power as: P = ρgA1 kt And thus: ∆h kf + kt 23 (3.17) 1 η = kt (kf ) 2 3 (kf + kt ) 2 (3.18) 34 Chapter 3. Extractible Power from a Single Channel This gives the extractible power as a function of the turbine resistance. See Figure 3.3. Equation (3.18) is then differentiated to determine the maximum resistance to be applied, such that: 1 − 1 (kt − 2kf ) · kf2 dη = 2 5 dkt (kt + kf ) 2 (3.19) Setting (3.19) to zero and solving for kt indicates that the maximum power that can be extracted from a tidal strait occurs when kt = 2kf . Taking this relationship between kt and kf and substituting back into (3.18) shows that the maximum extraction efficiency is 2 . 33/2 The potential power is then given as: P = 0.38ρg∆hQ (3.20) At this maximum, the change on the flow velocity may be computed such that p kf 1 Qreduced =√ =p Q kt + kf 3 (3.21) at maximum extraction, the flow will be reduced to 57.7% of natural. 3.4 Approximation of Sinusoidal Forcing Up to this point the analysis has been considering a fixed tidal head across the channel; Equation (3.15) or (3.20) may be evaluated for any time series of ∆h and Q depending on the drag law being assumed. The average power may be more easily obtained, however, if it is assumed ∆h varies 35 Chapter 3. Extractible Power from a Single Channel in time as: ∆h(t) = ∆hmax sin(ωt) The average power as a fraction of peak power may then be computed from Equation (3.12) as R 2π P̄ ω Pmax 0 = (| sin(ωt)|)2 dt 2π ω = 0.5 (3.22) = 0.556 (3.23) for linear drag or from Equation (3.17) P̄ Pmax R 2π ω = 0 3 (| sin(ωt)|) 2 dt 2π ω for quadratic drag. Combining Equations (3.15) and (3.22) or Equations (3.20) and (3.23) gives: Linear Relation Quadratic Relation P̄ = 0.125ρg∆hmax Qmax P̄ = 0.221ρg∆hmax Qmax (3.24) as the average power over a complete tidal cycle, assuming sinusoidal forcing. This agrees exactly with Garrett and Cummins (2005) for a friction dominated quadratic drag-law flow. 36 Chapter 3. Extractible Power from a Single Channel 3.5 Variation in Turbine Behaviour The simple relationships that are developed in the above analysis largely stem from the assumption that turbine drag is either linear or quadratic with velocity. The governing equations and assumptions, however, may be applied to more complex turbine behaviour, provided that the timescale for changes the turbine is sufficiently small such that ht may continue to be given as a function only of velocity. The complexity of turbine resistance relationships is a result largely of the torsional load placed on the turbine itself. In this context, that load would be from a generator and would therefore be dependent on the electrical characteristics of the generator and any electrical systems connected to it. Due to this inter-relation with the electrical system, more complex modeling of turbine resistance is outside of the context of this thesis. As such, a general solution will be presented. As before, power is given by: P = ρgu1 A1 ht (u1 ) (3.25) The difficulty in forming a generic relationship is determining flow velocity as a function of turbine (combined with natural) resistance. Obviously either through analytic or iterative means, it is requisite that one be 37 Chapter 3. Extractible Power from a Single Channel able to determine u1 (ht , hf , ∆h) such that: ∆h = ht (u1 ) + hf (u1 ) (3.26) Thus Equation (3.25) may restated as: P = ρgu1 (ht , hf , ∆h)A1 ht (u1 , ∆h, hf ) (3.27) and solves for an arbitrary function of ht and hf . 3.6 Characteristic Region - British Columbia’s Inside Passage British Columbia’s Inside Passage is the body of water between Vancouver Island and the mainland of British Columbia, comprised of the Johnstone, Georgia and Juan de Fuca Straits, see Figure 3.4. It was identified in the Canadian Ocean Energy Atlas (using kinetic energy flux method) to be able produce on order of 4GW of power (Tarbotton and Larson (2006)) Utilising the methodology developed in Section 3.3, the extractable power will be estimated. The Canadian Department of Fisheries and Oceans (DFO) maintains water level and current data for the inside passage. Using water levels from Victoria and Port Hardy, BC an appropriate ∆h can be determined 38 Chapter 3. Extractible Power from a Single Channel Figure 3.4: Inside Passage with measurement locations. Image source: NASA Visible Earth from the difference in tide levels. A subset of data used is shown in Figure 3.5. DFO current data set from station J09 (50 ◦ 22.400000 N 125 ◦ 39.500000 W) includes flow speed and direction for depths of 30m, 86m and 225m (total depth of 264m). The velocity component parallel with the channel direc39 Chapter 3. Extractible Power from a Single Channel !h (m) 2 !h Flow Rate 500 400 1.5 300 1 200 0.5 100 0 0 -0.5 -100 -1 -200 -1.5 -300 -2 02-Feb-77 04-Feb-77 06-Feb-77 Flow Rate (103 m3/s) 2.5 -400 08-Feb-77 Time Figure 3.5: ∆h between Victoria, BC and Port Hardy, BC and u at Station J09 (50 ◦ 22.400000 N 125 ◦ 39.500000 W) as a function of time (subset of data) tion was integrated over the cross sectional area of the flow; linear interpolation is used between the data points, see Figure 3.6. It was assumed that the top 30m had uniform velocity and the bottom was subject to no-slip. It was found that there was a small phase-shift between the velocity and the head difference. The tidal range was shifted by approximately 45 minutes to correct this. In the model developed, it was assumed that the acceleration of the fluid may be neglected. The very small phase shift in the observed data suggests that this assumption is valid for the Inside Passage. 40 Chapter 3. Extractible Power from a Single Channel Velocity 0 Depth (m) 50 100 150 200 250 Figure 3.6: Assumed characteristic velocity distribution over depth at DFO station J09 As discussed above, there is some ambiguity in the analysis as to the appropriate ∆h versus velocity relationship. For the case of the Inside Passage, this relationship can be determined through analysis of the observed data. By comparing ∆h to Q (and therefore also to u) for the phase corrected data, it is shown that there is a significantly stronger relationship between h ∝ Q (Figure 3.7) than h ∝ Q2 (Figure 3.8). The linear relation has a correlation coefficient of r2 = 0.97 while r2 = 0.9 for the quadratic relation. Similar observations have been made by Mattsson (1995) who sug41 Chapter 3. Extractible Power from a Single Channel gest that the linear term may be a function of rotation of the Earth. Given that the linear correlation is counter to typical quadratic drag, evaluation of this function should be performed on a situation-by-situation basis. For the remainder of this assessment, the system will be modeled using linear drag. 2.5 2 1.5 z0-z2 (m) 1 0.5 0 -0.5 R2 = 0.9682 -1 -1.5 -2 -400000 -300000 -200000 -100000 0 100000 200000 300000 400000 500000 Flow Rate (m3/s) Figure 3.7: Correlation of head loss with volumetric flow rate in BC’s Inside Passage Using Equation (3.15) applied at each point of the velocity data set and the two water level data sets, the extractable power can be computed (Figure 3.9). This yields an average power output of 477MW, a relatively small (though not insignificant) value. 477MW is approximately 5% of the in42 Chapter 3. Extractible Power from a Single Channel 4 3 z0-z2 (m) 2 1 0 -1 -2 -1.5E+11 -1E+11 -5E+10 0 5E+10 1E+11 1.5E+11 2E+11 2.5E+11 Square of Flow (m6/s2) Figure 3.8: Correlation of head loss with the square of volumetric flow rate in BC’s Inside Passage 43 Chapter 3. Extractible Power from a Single Channel stalled generation capacity of British Columbia (Ministry of Energy, Mines and Petroleum Resources, 2007) It should be noted that this value is just 13% of that predicted by the kinetic energy flux model (Tarbotton and Larson, 2006). As such, a channel that was previously thought to be very energy rich actually has a relatively small generation capacity. 2500 Power (MW) 2000 1500 1000 500 0 02-Feb-77 03-Feb-77 04-Feb-77 05-Feb-77 06-Feb-77 07-Feb-77 Time Figure 3.9: Power as a function of time in BC’s Inside Passage (subset of data) 3.7 Costs of Production An important limitation on the development of tidal energy is the cost of fabrication and installation of the turbines. To reduce costs, developers 44 Chapter 3. Extractible Power from a Single Channel would be wise to install the turbines in the energetic regions as the power that may be extracted by a turbine is proportional to the kinetic energy (KE) flux through the swept-area of the turbine; the higher the KE flux, the less swept area is required per unit of generation.1 Recall that KE flux is given by: ˙ = 1 ρAt u3 KE t 2 (3.28) This relationship between KE flux and the cube of velocity has a dramatic effect on generation capacity is installed. Given that power and velocity have been computed as a function of turbine resistance, the required swept area may be given by: At = P 1 ρu3t 2 (3.29) The results of combining Equation (3.29) with Equations (3.5), (3.12), (3.7) and (3.17) are shown in Figure 3.10. If it is assumed that to add swept area, one simply installs additional turbines and if installation costs of a single turbine remain constant (ie. discounting economies of scale), the cost of production can rise as fast as the cube of the installed capacity (for linear drag) as a result of the decrease in power density. Costs of generation 1 It is very important to note the difference between the use of corrected kinetic energy flux through the swept area of the turbine and the natural kinetic energy flux through the cross section of the channel, even if the areas are equivalent. The use of the natural flux is not corrected to the resistive effects that the installation of turbines will cause and thus will be reduced upon installation. 45 Chapter 3. Extractible Power from a Single Channel have been estimated to be approximate £0.057/kWh ($0.116/kWh) (Royal Academy of Engineering, 2004) for initial in flows with the natural energy density. Given these rates are already roughly double current electricity prices, the increasing cost with installed capacity may make utility scale generation prohibitively expensive. 50 Increase of Cost per W Production 45 Linear Drag Quadratic Drag 40 35 30 25 20 Maximum Production 15 10 5 0 0 0.5 1 1.5 2 2.5 kt / kf Figure 3.10: Required increase in turbine swept area per unit of power production as a function of installed turbine capacity. 46 Chapter 4 Extractible Power from a Split Channel The limits of energy extraction in a simple tidal channel were addressed in Chapter 3. An extension of this analysis is to consider flow through a split channel or around an island. This situation could arise where turbines are to be installed in a region with shipping traffic and it is deemed unsafe to install turbines in navigable waters. As a remedy, turbines would be installed in one sub-channel, leaving the other free for navigation. Like the situation in Chapter 3, the channel is connected at each end to an ocean and subject to hydrostatic tidal forcing. Again, the oceans are sufficiently large that they are not influenced by the dynamics of the channel in question. It is assumed that the length of the channel is small compared to the wavelength of the tidal forcing. At the centre of the channel lies an island, creating two sub-channels. Each sub-channel has the same depth as the main channel. A representative channel arrangement is shown in Figure 4.1. 47 Chapter 4. Extractible Power from a Split Channel (1-r)Q Q Ocean Ocean z0 LF rQ Lu LI Ld Figure 4.1: A tidal channel connected to two infinite oceans; an island separates the channel and tidal energy conversion devices are installed in one sub-channel. 4.1 Head Loss Applying Bernoulli’s theorum across the channel gives: 0 0 0 0 p0 u20 p2 u22 + + z0 − hu − hI − hd = + + z2 ρg ρg 2g 2g (4.1a) 0 0 0 0 p0 u20 p2 u22 + + z0 − hu − hF − hd = + + z2 ρg ρg 2g 2g (4.1b) where u is velocity, h is head loss, p is pressure, z is surface elevation, ρ is fluid density and g is gravitational accretion. Subscripts 0, u, I, F , d and 2 correspond to the upstream ocean, upstream combined channel, impeded channel, free channel, downstream combined channel and downstream ocean respectively. 48 z2 Chapter 4. Extractible Power from a Split Channel Unlike the analysis of the previous chapter, only quadratic drag will be considered as the channels involved will typically be shorter and thus entrance and exit effects will presumably become dominant; these head losses are strongly correlated to u2 . As such, head loss will be as: h = ku2 where k is a coefficient that characterizes the frictional or turbine resistance of the channel. The head difference, ∆h, across the entire channel can then be expressed as: For the impeded channel (with turbines) ∆h = z0 − z2 = ku + r2 (kI + kT ) + kd + kex Q2 (4.2a) For the free channel (without turbines) ∆h = z0 − z2 = ku + (1 − r)2 kF + kd + kex Q2 (4.2b) where kT , kI , kF , ku , kd and kex refer to the turbine, impeded channel, free channel, upstream, downstream and exit head loss coefficients respectively; Q is the total volumetric flow rate through the channel and r is the fraction of the total flow passing through impeded channel, obtained by 49 Chapter 4. Extractible Power from a Split Channel equating Equations (4.2b) and (4.2a) as: r= 1 √ 1+ α+β (4.3) α= kT kI β= kF kF (4.4) where Given that a fluid’s ability to do work can be expressed as: P = ρgQ∆h (4.5) extraction efficiency, η, can be computed as ratio of the power extracted by the turbines and the total fluid power of the channel, using Equations (4.5) and (4.2a): √ α(1 + α + β) Q ∆hT · rQ √ = η= 2 ∆h · Q0 α + β + γ(1 + α + β) Q0 (4.6) where ∆hT is the head difference across the turbines and Q0 is the natural flow rate without any energy extraction and γ is defined as: γ= ku + kd + kex . kF Q/Q0 is often close to unity and is left as a separate term as the reduction 50 Chapter 4. Extractible Power from a Split Channel of total flow is usually quite small; it can be expressed as: γ + r2 β Q = Q0 γ + r2 (α + β) (4.7) Computation of Equation (4.6) is shown in Figure 4.2. As can be seen, the maximum extraction efficiency occurs at a relatively low value of α and as β approaches zero. 5 0.025 4 " 3 0.03 2 1 0.035 0 0 5 10 15 20 ! 25 30 35 40 Figure 4.2: Extraction efficiency for various values of α and β where γ = 3. 4.2 Characteristic Region - Current Passage, British Columbia As a case study, Equation (4.6) is applied to Current Passage in Johnstone Strait, British Columbia (Figure 4.3). 51 Chapter 4. Extractible Power from a Split Channel 4.2.1 Estimation of Bottom Friction In order to determine α, β and γ, an estimate of the head loss due to bottom friction must be made. For the purposes of this analysis, ∆h over a reach is given by: ∆h = f L Q2 L V2 =f Rh 2g Rh 2gA2 (4.8) where L is the channel length, Rh is the hydraulic radius of the flow, V is the average velocity and f is a bottom roughness coefficient. k can then be defined as: k= 4.2.2 fL Rh 2gA2 (4.9) Resource Inventory The passage in question has a peak tidal range of 2.1m, a peak flow of 325·103 m3 /s, Lu = 6.5km, Ld = 12.5km and LI , LF = 4km, Au = 214·103 m2 , Ad = 200 · 103 m2 , AI , AF = 120 · 103 m2 , Rhu , Rhd = 75m, RhI , RhF = 72m and estimated f = 2.35 · 10−4 . Therefore: β ≈ 1.0 γ ≈ 2.6 The computation of Equation (4.6) for the senario in question is shown in Figure 4.4. There is an upper limit to the amount of power that can be extracted 52 Chapter 4. Extractible Power from a Split Channel Figure 4.3: Current Passage in Johnstone Strait Extraction Efficiency " 0.04 0.03 0.02 0.01 0 0 10 20 30 40 50 ! Figure 4.4: Extraction Efficiency vs. α for Current Passage, British Columbia, a split tidal channel. β ≈ 1.0, γ ≈ 2.6 53 Chapter 4. Extractible Power from a Split Channel as increasing α both reduces the total flow and diverts flow through free channel. Therefore as α → ∞, ∆hT → ∆h and rQ → 0 yielding η → 0. As additional turbine resistance is applied, the flow in the impeded channel reduces as the head across the turbines is increased. It should be noted that the majority of the effect of increasing turbine resistance is not to decrease the total flow, but to divert the flow to the free channel. See Figure 4.5. Fraction of Natural Flow 1 0.8 Q r 0.6 0.4 0.2 0 10 20 30 40 50 ! Figure 4.5: Changes in total and channel flow rates as a function of turbine resistance α. Knowing the natural flow rate and bathymetry of Current Passage, the peak natural kinetic energy flux through the impeded channel is 1.975 · 109 W . However, the peak maximum extractable power can be expressed as: P = ηρgQ∆h = 261 · 106 W (4.10) 54 Chapter 4. Extractible Power from a Split Channel The maximal extractible power is only 3.8% of the fluid power (ρgQ∆h) of the entire channel system. This is 13% of the resource that was identified by Tarbotton and Larson (2006) using the kinetic energy flux method (P = 1 ρAu3 ). 2 Also to note, as either f , the bottom roughness parameter or LF , the length of the free channel, approach zero, the flow is increasingly able to divert to the free channel and r will approach zero for any non-zero value of kT . As such, the extractable power will also fall to zero. This is especially important as high energy flows often occur in regions with little impedance. The bottom roughness coefficient effects both the natural kinetic energy flux and the maximum extractable power. As such, the locations with the highest natural kinetic energy flux may not be optimal locations for the installation of tidal energy conversion devices. 55 Chapter 5 Conclusions and Recommendations 5.1 Conclusions Tidal-in-stream power is one of many techniques that are currently being proposed in the search for replacements for fossil fuel in meeting our energy needs. While extraction devices are being designed, little research has been conducted as to the limits of energy extraction from natural tidal flows. The investigation of these limits is critical as an inaccurate inventory of the available resource could have dramatic impacts, skewing both financial and energy planning. Garrett and Cummins (2004) and Garrett and Cummins (2005) demonstrate that the most commonly used metric for evaluating the tidal resource, kinetic energy flux, is unreliable. This methodology does not account for the increased resistance due to the turbines and the corresponding effect on the total flow. The motivation for using the kinetic energy flux approach lies in the analogy drawn between tidal and wind power; 56 Chapter 5. Conclusions and Recommendations the momentum deficit of the wake of a wind turbine can be dissipated and the turbine’s effects are not felt outside of the near-field. Given that the kinetic energy flux is not a reliable metric, new methodology must be developed. In any accurate resource assessment, it is critical to understand the physics that drives a tidal flow: a finite (typically small) head difference across a channel drives the flow which is retarded by friction (and potentially inertia); the interplay between driving and resistive forces determines the flow rate. If energy extraction devices (turbines) are installed, they will act as an additional resistive force, reducing the flow. Given that power production is the product of head difference across, and the flow through a turbine, too little resistance will not generate a sufficient pressure differential while too much resistance will choke the flow. An ideal turbine resistance therefore exists that will allow maximum power extraction. The above ideal amount of turbine resistance has been determined to be a function of the natural parameters of the channel. It has been demonstrated through fundamental fluid mechanics principles that for quadratic drag in a single channel, maximum power will be produced if the turbine resistance is exactly twice the natural resistance of the channel. At this point, 38% of the natural fluid power of the system is available to be extracted. This results in a reduction of the flow to 58% of natural. These figures are smaller if the flow behaves according to the linear drag model. In 57 Chapter 5. Conclusions and Recommendations this situation, the maximum production is 25% of the natural fluid power and the flow is reduced to 50% of natural. For the situation where the tidal channel splits into two sub-channels in which turbines are installed in one, the available power drops significantly below the single channel case as a result of flow diversion from the impeded channel to the free channel. The extractible power is a function of the tidal range and natural flow rate, combined with the relative resistances of the two channels. It is important to note that these limits relate to the total amount of power taken from the flow, not the amount of power extracted by the turbines. For example, drag on the supports of a turbine will remove energy from the flow but will not add to the power produced. The limits also do not account for efficiency of the turbine or generation systems. Given that no device may have an efficiency of unity, extracted power will always be lower than these limits. Implicit in the previous analysis is that these are absolute limits based on the flow physics, no amount of technology development will result in an increased production capacity (for flows where the assumptions and boundary conditions hold). Applying the model developed in this thesis to the Inside Passage of British Columbia indicates a total extraction of 477MW is possible: the available power in a large region identified to be energetic has a small but not insignificant production capacity (approximately 5% of British Columbia’s total installed generation capacity). Previous studies using the kinetic en58 Chapter 5. Conclusions and Recommendations ergy flux have overestimated the available resource by approximately a factor of seven. Analysis was also performed on Current Passage, a small region involving split tidal channels, considering the situation if turbines were installed in only one channel. Due to diversion of the flow from the impeded channel to the free channel, only 3.8% of the fluid power of the strait is extractable. 5.2 Limitations The analysis conducted has neglected the effects of certain physical phenomenon that may have an influence under some conditions: • Acceleration terms are not accounted for in the head drop model. Inertia will tend to resist the change in flow velocity, thus for long and deep channels with rapidly varying tidal levels, the inertial terms in the momentum equation may become important. As such, for flows where inertia is dominant, this model may over estimate the resource. The acceleration term is accounted for in the model developed by Garrett and Cummins (2005). • If the period of a shallow water wave traveling through the channel under investigation is not significantly smaller than the period of tidal forcing, resonant effects may become important. The model 59 Chapter 5. Conclusions and Recommendations assumes that the flow is quasi steady-state and thus could not be applied if resonance is important. The purpose of this model is to provide an accessible alternative to the kinetic energy flux for the assessment of a tidal-in-stream energy resource. If the limitations of the model are understood and it is not applied where the assumptions and boundary conditions are invalid, this model can serve as a useful tool for energy planners, policy makers and technology developers. 5.3 Future Work As a result of this investigation into the limits of resource extraction, several areas of research are considered lacking. As such, additional research should be conducted in the following areas: • An important and relatively common phenomenon in tidal flows is the effect of resonance. In these situations, the strait could not be considered steady-state at any particular time as the flow is dependent on conditions at other points in the tidal cycle. The implications of this resonance are not understood in the context of power generation. • The head loss across a turbine in a typical operating condition is not readily available. This is largely due to the rotational speed, and 60 Chapter 5. Conclusions and Recommendations therefore resistance of a turbine is not solely a function of the fluid dynamics and is in some effect determined by the electrical characteristics of the generator. An improved model of this behaviour would provide a more realistic tidal energy inventory. • An explanation of the apparent linear drag in the Inside Passage (and other locations) is needed. As a general characteristic, investigation into the prominence of linear versus quadratic drag could allow for easier ‘rough estimates’ of a resource. In conclusion, the assessment of a tidal-in-stream resource may be determined in many situations through the use of basic natural flow conditions as inputs to the model developed in this thesis. This model is a replacement for the use of kinetic energy flux, as this KE flux is not a reliable metric in evaluating the resource; it tends to significantly overestimate the available power. As such, previous inventories that use the kinetic energy flux are likely not accurate. Beyond the physical limitations of power extraction, significant economic challenges exist at high rates of extraction; the velocity of the flow reduces significantly as peak production is approached, which results in reduced power density and thus requiring additional turbine area per unit of power produced. 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