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| Fig. 1: The Bay of Fundy (North East) and Gulf of Maine (South West) Tidal system. Minas Passage connects the lower right finger above the Bay of Fundy, to the main body of water. (Source: Wikimedia Commons). |
The rotational and gravitational dynamics of the Earth-Moon-Sun system have wonderful repercussions for life on Earth, one of which is the oscillation of water in the ocean, or the tides. Kinetic and potential forms of energy associated with the ocean tides are introduced in previous works. [1,2] Unlike wind and solar energy, tidal currents are precisely consistent, and with the high density of seawater, tidal currents carry with them a lot of energy.
Of particular interest in this report is the Bay of Fundy Gulf of Maine tidal system that is being forced near resonance to give extremely large tidal ranges. [3,4] Southern Nova Scotia is located on the bay of Fundy (Fig. 1). In parts of the Bay of Fundy, such as Minas Basin, which is connected to the bay by a narrow channel called Minas Channel, record-setting tidal exchanges in excess of 50 feet have occurred. [3] Consequently, the Bay of Fundy is a hot-zone when considering harnessing energy from this impressive tidal resonance system. Both potential energy extraction using tidal barrages (Fig. 2), and kinetic energy extraction using turbines (Fig. 3), have been considered to tap into this tidal energy. [5]
To make a realistic estimate of the amount of energy that could be extracted from the Bay of Fundy tides, the concept of tidal current energy will be utilized. Unlike tidal potential energy, tidal current energy does not require damming a tidal estuary. It relies instead on the use of tidal turbines. As a result it is less geographically constrained than tidal potential systems.
The system postulated in this analysis is a string of turbines in a row, all in the same plane perpendicular to the current, also known as a turbine fence, or tidal fence. A visual representation of depth-averaged tidal currents through Minas Passage with and without turbines is provided by R. P. Mulligan et al. in their 2014 report. [4] In a more detailed follow-up of the same study, Ashall et al. conducted analysis to inform their detailed study of the impacts of a turbine farm on suspended sediment concentration due to tidal turbines and further visual representations of the flow through Minas passage can be found there. [5] Rather than deploy a turbine farm however, the present study deploys a single linear array of turbines across the passage.
The maximum power available in a tidal current is given by Eq. (1). This is simply the kinetic energy flux through a given area, where A is the cross-sectional area of the channel, and v is the undisturbed, average, free stream velocity.
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(1) |
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| Fig. 2: The Tidal Barrage of the Rance tidal plant, on the Estuary of the Rance River, in Bretagne, France. The facilty is used to store and then harness the potential energy of the tides. (Source: Wikimedia Commons). |
From Eq. (1) it is clear that the available power is a function of velocity cubed, so a fewer quantity of turbines will be required for same power generation if a region with faster currents is used. By conservation of mass, the fastest currents occur where the channel is the narrowest. This is why Minas Passage is used. It is a particularly narrow region (4.5 km wide), where tidal currents rush past Cape Split, NS, during an ebb tide between 5 and 8 knots. [4,6] An average speed of 3.3 m/s will be used in this analysis as the peak ebb-tide current. Additionally, this analysis will consider turbine energy generation on an ebb tide only. Eq. (1) provides the available power in the flow (this is just the kinetic energy flux of water through Minas Passage), which is dependant on an assumption of the depth and width of the passage.
However, turbines actually create a pressure loss. In an ideal turbine, the free stream current slows from a speed of v, to 2/3 v in the turbine, to 1/3 v at the turbines exit, as shown by the stream tube derivation. [7] The energy losses created by a single, isolated turbine obstructing a flow are derived using the continuity, conservation of energy, and momentum equations in Garrett and Cummins. [7] The result is that the maximum amount of energy that can be extracted by a single turbine is 59% (16/27) of the maximum kinetic energy flux. This limit is for an ideal turbine and is known as the Betz-Lancaster Limit. The resulting maximum power per turbine is thus given by Eq. (2), where the term η denotes the efficiency of conversion from kinetic to electrical energy within the turbine itself.
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(2) |
This result from Eq. (2) cannot simply be multiplied by the number of turbines we can fit across Minas Passage. This is because, in reality, the fact that Minas Passage is a finite channel with a limited width requires deeper analysis. In their paper in 2007, Garret and Cummins go over this analysis in excellent detail. [7] Ultimately, as we apply in this report, when the total cross-sectional area of all turbines is considered with respect to the total cross-sectional area of the channel, the channel boundaries are seen to confine the flow and increase the total pressure drop that can occur, therefore increasing maximum power extracted by a turbine fence above the Lancaster-Betz limit for an isolated turbine. [7] This effect is captured by area ratio of the total turbine area to the channel area in the extra parameter shown in Eq. (3).
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(3) |
Of course, by adding too many turbines in the channel the current speed will reduce and become too slow, choking the flow and reducing the power extracted. [8] There are many good discussions of optimizing number of turbines in the flow to maximize power output. [7,9] Optimizing for peak power extraction will not be tackled here, rather, a rough estimate of power extraction for turbine fence will be made based on the assumptions listed in Table 1 below.
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| Fig. 3: SeaGen, the second generation tidal turbine prototype, installed in Ireland by Marine Current Turbines capable of generating 1.2 MW for roughly 20 hours per day to the grid between 2008 and 2019. (Source: Wikimedia Commons). |
Using the assumptions listed in Table 1, the following intermediate numbers can be calculated:
Number of turbines that fit across channel: 253 Turbines
Total area of turbines: 44,731 m2
Total area of passage: 270,000 m2
Ratio of turbine area to total area: 0.166
Finally, using Eq. (3), the maximum power that can be extracted with this configuration of turbine fence is computed as
| Pmax (all turbines) | = | 16 27 |
× (1 - | 44,731 m2 270,000 m2 |
)-2 × | 1 2 |
× 1025 kg m-3 × 44,731 m2 × (3.3 m s-1)3 × 0.6 |
| = | 4.2 × 108 W | ||||||
Of course, this power can only occur when the ebb flow is 3.3 m/s. Refined analyses compute this power over the range of all flow rates during a tidal cycle and average it out. But if one looks at the tide charts for Minas Basin, the slope of water level vs time is relatively constant for approximately 4 hours. So by combining a 4 hour ebb, with a tidal oscillation period of 12.42 hours, we get 7.73 hours of ebbing tide per day. In other words, 0.44 GW can be generated for 7.73 hours per day. As a sanity check, Karsten et al. conduct a very comprehensive mathematical model and report that 0.77 GW of power can be extracted, while only disturbing the amplitude of tidal resonance for the Bay of Fundy / Gulf of Maine system on the order of 1%. [8] This is in the same ballpark as our current analysis.
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| Table 1: Assumptions made to facilitate the estimates calculated in this report. |
With our current tidal fence in Minas Passage generating 0.44 GW for 7.7 hours per day, the total energy generated in a year by the system is:
| E | = | 4.2 × 108 W × 3600 sec h-1 × 7.7 h d-1 × 365 d | = | 4.25 × 1015 J |
To put this analysis in perspective, we can go back and look at the annual energy demand in Nova Scotia. The province of Nova Scotia is heavily reliant on coal and coke products to produce as much as 50% of its electricity. Thus the capacity for a turbine fence in Minas passage to help alleviate this coal dependence is an interesting question. In 2019, Nova Scotia energy demand was roughly 175 PetaJoules (1.75 × 1017 J). Therefore, the percent of the 2019 Nova Scotia energy demand that could be provided for by the tidal fence in this analysis is 4.2 × 1015 J / 1.75 × 1017 J ~ 2.4 %.
We can also look at yearly average power instead by taking the total yearly energy demand in Joules and dividing by the amount of time in a year. We find that Nova Scotia energy demand averages to 5.5 GW. This is a yearly average. The yearly average power production for our ~0.5 GW system is computed as 4.45 × 1015 J y-1 / (31.53 × 109 s y-1 = 0.14 GW.
However, it has been noted that our current system is not optimized. Other studies have optimized turbine placement and size and to predict maximum power generation capacities about 10x higher. For example, Karsten et al. reported 6.95 GW (this is instantaneous, not a yearly average). [8] However, rates of power extraction as high as these estimates have been predicted to cause very serious changes to the tidal amplitude experienced by the Bay of Fundy / Gulf of Maine resonant tidal system.
Karsten et. al. also found that if operating at peak extraction, around 7 GW from Minas Passage, the tidal amplitude of the Bay of Fundy Gulf of Maine system would change by 30-40%. [8] In fact, at 6.95 GW, a ~35% drop in tidal amplitude in Minas Basin, and a ~15% rise in tidal amplitude in the coupled, Gulf of Maine system would be observed. They also note that the inverse effect could occur (increasing tidal amplitudes by up to 30%) by adding a tidal barrage in Minas Passage, pushing the system closer to resonance. [8] For tidal turbines, other researchers have reported an expected 23% reduction in the Bay of Fundy tidal amplitudes for 6 GW power extraction, or 14% reduction in amplitudes at 5.0 GW power extraction. [4] Ashall et al. conducted a more detailed study in 2016 of the impacts of a turbine farm would have on on suspended sediment concentrations, reporting a reduction in suspended sediment by up to 37% at maximum power extraction, causing both physical and biological changes to the Minas Basin. [5]
This analysis has found that while the concept of harnessing tidal energy from systems near resonance appears very attractive on the surface when spring tides bring tidal ranges on the order of 40-50 feet, tidal oscillations in the Bay of Fundy, (and around the world), are a dynamic system and it is important to consider the far-field implications of extracting energy from such systems at large scales. In addition to changes in tidal amplitudes on the order of tens of a percent, the construction of tidal energy systems will create step changes in that ecosystem that include changes in flow rates, sediment transport, species mobility, erosion, and other ecological considerations that muddy the waters of those wishing to harness the energy of tides in places like the Bay of Fundy.
© Gibson Clark. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] N. Peterson, "Sun, Moon, Oceans: The Potential of Ocean Tidal Energy," Physics 240, Stanford University, Fall 2018.
[2] J. Wilson, "Tidal Power - Renewable Energy from the Ocean's Movement," Physics 240, Stanford University, Fall 2021.
[3] C. Garrett, "Tidal Resonance in the Bay of Fundy and Gulf of Maine," Nature 238, 441 (1972).
[4] R. P. Mulligan et al., "Effects of Tidal Power Generation on Hydrodynamics and Sediment Processes in the Upper Bay of Fundy," 48th Specialty Conf. on Coastal, Estuary and Offshore Engineering, 29 May 13.
[5] L. M. Ashall, R. P. Mulligan, and B. A. Law, "Variability in Suspended Sediment Concentration in the Minas Basin, Bay of Fundy, and Implications For Changes Due to Tidal Power Extraction," Coast. Eng. 107, 102 (2016).
[6] B. J. Todd and J. Shaw, "Discovering the Seafloor of the Bay of Fundy," in Voyage of Discovery: Fifty Years of Marine Research at Canada's Bedford Institute of Oceanography, ed. by D.N. Nettleship (Bio-Oceans Association, 2014), p. 231.
[7] C. Garrett and P. Cummins, "The Efficiency of a Turbine in a Tidal Channel," J. Fluid Mech. 588, 243 (2007).
[8] R. H. Karsten et al., "Assessment of Tidal Current Energy in the Minas Passage, Bay of Fundy," Proc. Inst. Mech. Eng. A-J. Pow. 222, 493 (2008).
[9] C. Garrett and P. Cummins, "The Power Potential of Tidal Currents in Channels," Proc. R. Soc. A 461, 2563 (2005).
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