## Abstract

For tidal power barrages, a breast-shot water wheel, with a hydraulic transmission, has significant advantages over a conventional Kaplan turbine. It is better suited to combined operations with pumping that maintain the tidal range upstream of the barrage (important in reducing the environmental impact), and is much less harmful to fish. It also does not require tapered entry and exit ducts, making the barrage much smaller and lighter, so that it can conveniently be built in steel. For the case of the Severn Estuary, UK, it is shown that a barrage at Porlock would generate an annual average power of 4 GW (i.e. 35 TWh yr^{−1}), maintain the existing tidal ranges upstream of it and reduce the tidal ranges downstream of it by only about 10%. The weight of steel required, in relation to the annual average power generated, compares very favourably with a recent offshore wind farm.

## 1. Introduction

Tidal barrages can produce more power, with less environmental impact, if they combine pumping with power production. This can readily be seen by considering the simple case of a small bay of area *S* opening into an infinite deep ocean with a tide *a*cos(*ωt*). Consider first a tidal barrage without pumping, producing a pressure difference directly proportional to the volume flow rate through it. This system is readily explored by means of the standard electrical analogy (e.g. [1], p. 104) of pressure with voltage, and volume flow rate with electrical current. On this analogy the bay becomes a capacitance of *C* *=* *S/ρg* ([1], p. 200, (3)) and the barrage becomes a resistance *R*_{B}, whose value we wish to optimize to give maximum power dissipation in the resistance. The equivalent circuit is shown in figure 1.

Also shown is the phasor (Argand) diagram of the voltages (pressures) across the capacitor and resistor, which are at right angles because the impedance of a capacitance *C* is 1/i*ωC*, and which sum to that of the tide outside the bay (assumed, because the bay is small, to have the same pressure amplitude *P* *=* *ρga* seen before the barrage was built). As the resistance of the resistor is increased, the voltage across it increases, but at the expense of the voltage across the capacitor, and thus the current. Since the two phasors are at right angles, point A in figure 1 describes a semicircle, as shown. The power dissipation in the resistor is the product of the voltage across it and the current, and is thus proportional to the hatched area in figure 1, which is at a maximum when the two sides of the triangle have equal voltages (pressures), as shown, of *ρga*/√2. It is then easy to show ([2], eqn (16)) that the average power is ¼*ρgSωa*^{2}*.*

In a neglected analysis of the Severn Barrage, UK, Robinson [3] extends such arguments to the case of a barrage represented not merely by a resistor *R*_{B} but by a resistor in series with an inductor *L*_{B}. Since the impedance of the inductor is i*ωL*_{B}, and thus of opposite sign to that of the capacitor, it can be used to reduce the overall impedance of the system, and thus increase the flow into the reservoir, and the power. As Robinson notes, the flow into the reservoir can in principle be made larger than it was without the barrage, increasing the tides from their pre-barrage values. He also notes, however, that this may cause unacceptable flooding (it certainly would in the case of the Severn Barrage, where the Somerset Levels are vulnerable) and he concludes ([3], p. 619) that a practical limit, worthy of future study, is to maintain the tidal range in the reservoir at its pre-barrage range in typical spring tides. In this paper, the simpler case of maintaining the pre-barrage tidal range exactly, in all tides, is considered. There may well be more power available, with still acceptable environmental impact, with more sophisticated strategies, but this point is not explored here.

Our simple case of a small bay is illustrated in figure 2. For simplicity the pressure across *R*_{B} has been maintained at its value in figure 1, but the tidal amplitude in the bay is now increased by a factor √2, to its pre-barrage value *a*. The current through the barrage (and thus the power, again represented by the hatched area) is thus also increased by a factor √2.

Since one of the major objections to tidal barrages is the environmental impact of the reduced tidal range in their reservoirs, this is a win–win situation: less environmental impact and more power.

However, we must consider the turbine characteristic needed to simulate a resistor in series with an inductor. The inductor in figure 2 has an impedance whose amplitude is (√2 – 1) = 0.414 times that of the resistor (in order to increase the current by √2), so their individual and combined pressure–flow characteristics, with a sinusoidal applied pressure *ρga *sin(*ω*t), are as shown in figure 3.

As the applied pressure completes a cycle, the flow through the turbine follows the red/green elliptical path. Where the path is red, the flow and pressure are of opposite signs, so the turbine is acting as a pump. This is very feasible from the electrical engineering point of view (a synchronous generator automatically becomes a synchronous motor when the applied torque changes sign) and has long been recognized as an option with tidal barrages ([4], §2).

It is necessary, however, to consider the energy required for pumping relative to that generated by turbine action. This is shown in figure 4, which plots the power output (before conversion losses) over a complete tidal cycle.

The negative powers, corresponding to pumping, are remarkably small—less than 4% of the positive powers corresponding to turbine action. There are three reasons for this:

— The impedance of the inductor is 0.414 times that of the resistor.

— If the current varies as cos(

*ωt*) then the power into the inductor varies as cos(*ωt*)sin(*ωt*) = ½sin(2*ωt*), whereas that into the resistor varies as cos^{2}(*ωt*) = ½(1 + cos(2*ωt*)). Even with equal impedances, the peaks of the latter are thus twice those of the former.— The peaks are 90° out of phase.

Because of the short duration of the pumping action, its energy (i.e. the area under the graph in figure 4) is relatively much less than that of the turbine action, only about 0.65% of it. The pumping energy input will be magnified by conversion losses, and the turbine energy output reduced by them, but if we assume an overall conversion efficiency of 64% (80% hydrodynamic, 80% hydraulic/electrical; see next section) the pumping energy input is still only about 1.6% of the turbine energy output. Thus, the scheme of figure 2 appears theoretically feasible.

There is a practical problem, however, with the Kaplan turbines conventionally ([4], §3) proposed for tidal barrages. At the points B in figure 3, there is pressure across the turbine, but no flow—a conventional turbine can only do this very inefficiently. This paper proposes instead the use of breast-shot water wheels, with a hydraulic transmission (figure 5)—the breast-shot water wheel features a curved ‘breast' with only a small clearance to the wheel blades, which thus act like the vanes in a vane pump rather than like the paddles is a ship's paddle-wheel. In particular, if the wheel stops, so does the flow, as required at points B in figure 3.

The breast-shot water wheel is actually a very old concept, dating from the early industrial revolution (there is an excellent example at Claverton, near Bath, UK, which was built in 1812 and is still working), but has attracted attention in recent years for small-scale hydroelectric applications in developing countries. Recent laboratory experiments are described in [5], where an efficiency of 80% was achieved over a wide range of flow conditions.

Apart from its ability to stop the flow efficiently, it has two other important advantages over a conventional Kaplan turbine:

— It can readily be given large blades to minimize the flow velocity through it. This means the tapered entry and exit ducts, conventionally ([4], §3) fitted to Kaplan turbines to minimize the turbine diameter, are not required. This makes the barrage much smaller and lighter—it can conveniently be built in steel rather than concrete.

— Fish can readily pass through it, in contrast to a Kaplan turbine, where the blade velocities are so high that they are very damaging to fish.

For the present application, the breast-shot water wheel needs to be set to run at any prescribed speed (and thus give any prescribed volume flow rate), whatever the pressure across it, so as to follow the characteristic in figure 3. This can be accomplished with a variable displacement hydraulic transmission, which acts like a gearbox with an infinitely variable gearing ratio. By coupling it to a synchronous generator, which runs at a constant speed set by the grid frequency, a prescribed volume flow rate can thus be achieved by suitable choice of the transmission gearing ratio, which will vary continuously over the tidal cycle.

## 2. Application to the Severn Barrage: power calculations

The Severn Estuary (figure 6) differs from the small bay envisaged earlier in two respects:

— It is large enough to have significant internal dynamics—the tidal range behind a barrage will increase as we move up the estuary, just as the tides presently do.

— The access to the ocean is impeded by the estuary further west, particularly for barrages further up the estuary.

Both these effects can be quantified using an analytical model of the tides in the Severn Estuary due to Taylor [6] and described in ([7], p. 276). This model was first applied to studies of the Severn Barrage by Robinson [3,8] and has independently been applied by the present author [9].

As described in [9], the two effects can be included by additional elements in the equivalent electrical circuit of figure 2; see figure 7.

The impedances *Z*_{1} and *Z*_{2} can be determined from Taylor's analysis, which begins ([6], eqn (1)) with the differential equation ([7], p. 274, eqn (4)) for tidal waves of elevation *η* in a channel whose depth *h* and width *b* both vary with distance *x* along the channel

If we seek harmonic solutions of angular frequency *ω* ( = 2*π*/12.4 h^{−1} for our case of the tides) of the form
*η*_{0}(*x*) is complex, then (2.1) reduces to

Taylor assumes that the channel tapers linearly in both width and depth from the end *x* *=* 0, and thus puts
*β* and *γ* are dimensionless constants. This reduces (2.3) to

From measurements at positions A–G in figure 6, Taylor takes *x* = 0 at G and estimates *γ* = 0.0003084 and thus *k* = 0.00655 km^{−1}. Note that *β* is immaterial—provided the width varies linearly from *x* = 0, it does not matter what the rate of variation is, because, for example, a wall could be added along the centre line of the channel, making two narrower channels, without affecting the tides.

Taylor now makes the substitution

After a final substitution *y* *=* 2*z*, this becomes
*H*_{1}^{(1)} and *H*_{1}^{(2)} are the first and second Hankel functions of order 1, representing tidal waves travelling east and west, respectively, and *K*_{1} and *K*_{2} are complex constants. These constants can be determined in a variety of ways.

(a) From prescribed tides (amplitude and phase) at each end of a finite length of channel (with

*x**>*0 at the eastern end). Two such complex numbers give two simultaneous equations for*K*_{1}and*K*_{2}*.*(b) From the (complex) ratio of pressure to volume flow rate (i.e. the channel impedance) at one end of the same finite length of channel, and a prescribed tide (amplitude and phase) at the other. This again gives two simultaneous equations for

*K*_{1}and*K*_{2}*.*This situation is considered, for example, by Robinson [3], when he establishes the downstream tidal disturbance produced by a barrage from the additional flow at the barrage and the channel impedance at his outer boundary (at his notional western channel end, west of A, figure 6), which he obtains from a numerical model of the ocean beyond it ([3], p. 621).(c) From a prescribed tide at the western channel end, and the assumption that the tides in the channel are a standing wave (i.e.

*K*_{1}=*K*_{2}, so that (2.9) becomes*ξ*=*KJ*_{1}(*y*), where*J*_{1}is a Bessel function of the first kind, of order 1). This is the case considered by Taylor, and he demonstrates that it correctly predicts the tidal variations at positions B–G east of his western channel end at A (figure 6).(d) From a prescribed tidal disturbance at the eastern end of a channel extending westwards to infinity, and the assumption that the tidal disturbance is a pure westward-travelling wave (i.e.

*K*_{1}*=*0). This is the Sommerfeld radiation condition, which applies very widely (e.g. [1], pp. 363–364), and is the assumption made in [9] in the author's analysis of the tidal disturbance produced by a barrage on its western side.

In this paper, we again follow procedure (d) for the case of a barrage at A. The channel to the west of A is assumed in [9] to double in width at A, and then to taper linearly from a notional apex at Abergavenny (figure 6). There is a different linear variation of depth with *x*, which gives a new parameter *k** = 0.00547 km^{−1} in place of *k* = 0.00655 km^{−1}. The complex pressure in the channel follows from (2.10), and so does the complex volume flow rate in it ([9], eqn (10)). Their ratio at A gives *Z*_{2} for a barrage at A and is given in table 1.

For barrages east of A, we need to consider the region between the barrage and A. Here we can follow procedure (b), because the channel impedance immediately west of A has just been calculated, and must equal that immediately to the east of it, so that the volume flow rates match when the pressures are the same. This matching condition leads to

The properties of this tidal wave, on both sides of A, are shown in figure 8.

The (complex) ratio of pressure to volume flow rate in this tidal wave gives us *Z*_{2} for positions B–E (table 1). They are the same as those obtained by the more roundabout procedure described in [9].

To obtain *Z*_{1} in figure 7, we need to consider the reservoir east of the barrage. As already noted in (c) above, Taylor's model, with his assumption of a standing wave, does fit measurements of the dynamic amplification of the tides in such a reservoir, from positions A to G. However, there are no tidal delays in such a standing wave, whereas in fact there is a delay of about an hour from position A to G. In [9] this delay is treated in an empirical way by keeping Taylor's expression *KJ*_{1} (*y*) for the tidal range, dividing the reservoir area into blocks between the positions A and G, and simply delaying the volume flow at the barrage associated with each block by the tidal delay at that block shown on the Admiralty charts. This delays the overall flow into the reservoir at the barrage, so that the reservoir is no longer equivalent to the capacitor shown in figures 1 and 2 but to a capacitor in series with a resistor *R*_{B}, as shown in figure 7. This resistor represents the energy losses through bottom friction, which are ignored in Taylor's standing wave, where volume flow rate and pressure are exactly 90° out of phase, as in figures 1 and 2.

It is shown in [9] that the inclusion of this resistor reduces the power from the barrage by 5–10%, and thus that the bottom friction in the reservoir is important to this extent. It is ignored west of the barrage, because the ratio of the energy dissipated by friction on a given bottom area to the kinetic energy in the water column above it falls in proportion to water depth. Thus, bottom friction will be less important in the deeper water west of the barrage.

In contrast to this empirical treatment of bottom friction, Robinson [3] treats bottom friction explicitly by an additional term in the differential equation (2.1). For present purposes, however, the simpler empirical treatment above is sufficient, with the tidal delays from the Admiralty charts now taken in typical tides, rather than the neap tide figures used in [9], which are appropriate to the reduced tidal ranges in the reservoir considered there. The values for the impedance *Z*_{1} obtained in this way are given in table 1.

With all the elements in the equivalent circuit of figure 7 thus defined, the power of the barrage can now be established by AC circuit theory. As in [9], we consider a pre-barrage tide of root mean square amplitude (the pressure *P* in figure 7) between spring and neap tides, in order that the power generated is representative of the annual average. This is a tidal amplitude of 4 m (i.e. ±4*ρ*g kPa pressure, 8 m tidal range) at section D. For each barrage location, we consider a range of barrage resistances (*R*_{B} in figure 7), and in each case adjust the barrage inductance (*L*_{B} in figure 7) to maintain the tidal amplitude east of the barrage at its pre-barrage amplitude—albeit that its phase has changed. The results are plotted in figure 9, where the plots are stopped at the point where the impedance of *L*_{B} reaches that of *R*_{B}, because beyond that the pumping losses will become significant by the arguments of the previous section.

A suitable choice for the peak head across the barrage is evidently 0.6 times the pre-barrage tidal amplitude (range/2). For this choice, the changes to the tides produced by the barrage are shown in table 2.

## 3. Outline design for the Nash Point/Hurlstone Point site

An attractive barrage site from the engineering point of view is mid-way between sections C and D, between Nash Point on the Welsh side and Hurlstone Point, near Porlock, on the English side (figure 6). Here the estuary narrows locally to 18 km width, and the seabed is level bare limestone, ideal for drilled pile foundations (in quarries similar rock is drilled, at typically 10 m h^{−1}, prior to pouring in explosive for blasting). The mean depth is close to 20 m [9], and a suitable width for the installation of breast-shot water wheels is 15 km—this width needs to be as large as possible to minimize the flow velocity through the turbines and thus the exit losses. The Taylor formula ((c) above) gives the maximum pre-barrage tidal range as 9.5 m, when the range at section D is at its maximum of 10 m. Thus, a suitable diameter for the hub of the wheel in figure 5 is 10 m, with its axis at mean sea level, so that it is never completely under or above the water level east of it. A suitable overall wheel diameter is 30 m, leaving 5 m clearance to the seabed, on average, to be spanned by the ‘breast' (figure 5). A suitable overall length of the wheel is 50 m, and, allowing 10 m for the bearings and their supports between wheels, this makes a total length of 60 m per wheel, giving 15 000/60 = 250 wheels in all. The wheels are fabricated in steel plate, like a ship, with extensive use of corrugation for plate stiffening, following modern shipbuilding practice (all the bulkheads in a modern tanker are corrugated). A single wheel is shown in figure 10.

The wheel runs on two roller bearings similar to those fitted to the turrets of some floating oil rigs (FPSOs). With the maximum water pressure of *ρg*0.6 × 9.5/2 = 28.7 KPa the force on each is 28.7 × 10 × 50/2 = 7.2 MN = 730 tonnes, which is well within their capabilities. The hydraulic power take-off is via hydraulic rams which operate in pairs like a strand jack, clamping alternately onto a brake disc on the wheel by means of automotive-style hydraulic calipers. The rams and calipers can be lifted clear of the brake discs, and the filler plates between wheels folded down, as shown in figure 11, so as to allow the wheel, with its bearings, to be floated into place or changed out for maintenance.

With rams of 600 mm bore and 300 mm rod diameter, the hydraulic pressures at the maximum water pressure come to 16.9 MPa pushing and 22.5 MPa pulling, which is well within conventional hydraulic practice. The very simple hydraulic design will give a high efficiency, so an overall wheel-to-electric efficiency of 80% is credible. Combined with the water wheel efficiency of 80% cited earlier [5], the overall conversion efficiency is 0.8 × 0.8 = 64%, so given the annual average tidal power of a little over 6 GW shown in figure 8 for this location (with the assumed head across the barrage of 0.6 times the pre-barrage tidal head), the annual average power of the barrage is 4 GW, in round figures. This corresponds to an annual energy output of (4 × 24 × 365)/1000 = 35 TWh yr^{−1}.

From table 1, the reservoir impedance at section D is √(12.76^{2} + 73.69^{2}) = 74.79 kPa/(Mm^{3} s^{−1}), so in a tide of average range 7.5 m the maximum volume flow rate at section D is
^{3} s^{−1}. The total flow cross-section of the wheels is 250 × 50 × 10 = 125 000 m^{2} = 0.125 Mm^{2}, so at our location the average flow velocity through the wheel in average tides is

This is exceptionally low for a tidal barrage—marine mammals, and indeed humans in kayaks, could pass through it with ease. Also, the average period of rotation of the wheel is 20*π*/3.45 = 18.2 s, so the number of stress cycles per year in the wheel, important for fatigue, is 365 × 24 × 3600/18.2 = 1.7 million, which is manageable. In appendix A, the thickness of the steel plates in the wheel are given, and the fatigue lives of the corrugation faces, the welds at their ends and the brake disc are shown to be acceptable. We are thus in a position to estimate the weight of steel in the wheel (table 3).

The cradle supporting the wheel bearings, and forming the ‘breast', is shown in figure 12. The corrugated components are not subject to significant cyclic loads, so are designed for sufficient strength, at the maximum pressure across the barrage of 28.7 kPa. The face width of the corrugations is 0.8 m, and its length between bulkheads is 10.5 m, from which the formulae in appendix A give stresses of 92 MPa and 86 MPa, respectively, which are acceptable. The weight of steel in the cradle and support structure (for the hydraulic rams; figure 10) is given in table 4, where the thickness of the curved ‘breast’ is estimated as 20 mm and the pile wall thickness as 30 mm. Thus, the total steel weight of a single 60 m unit of the barrage comes to 1600 + 900 = 2500 tonnes. This can be compared with a single unit of the Beatrice offshore wind farm north of Inverness, UK, which is currently under construction and whose steel weights are given in table 5.

Thus the units of the barrage have approximately 50% more steel in them than a unit of the wind farm, but they are approximately five times more powerful on average (4000/250 = 16 MW annual average power compared with about 3 MW annual average power from the 7 MW turbines on the Beatrice wind farm). Thus, the barrage uses less than one-third as much steel, per TWh yr^{−1}, as the wind farm.

## Data accessibility

This work does not have any experimental data.

## Competing interests

I have no competing interests.

## Funding

This work has been funded entirely by Rod Rainey & Associates Ltd.

## Acknowledgements

The author is pleased to acknowledge helpful conversations with Dr Richard Yemm, Dr Rick Jefferys and Mr Chris Binnie.

## Appendix A. Fatigue calculations on the water wheel

To calculate the fatigue life of the wheel, it is necessary to establish a representative pressure difference across the wheel (and thus the stress range in the blades as the wheel rotates). Since long-term fatigue damage is proportional to the fifth power of the stress range ([10], fig. 2–8), and the rotational speed of the wheel is also approximately proportional to the stress range, the rate of fatigue damage is proportional to the sixth power of the stress range. The representative pressure difference is thus the sixth root of the mean sixth power of the pressure across the wheel, as it varies over a 12.4 h tidal cycle and a 14.8 day spring/neap cycle. This is 69% of the maximum pressure range on the wheel blades, which is the maximum pressure across the barrage of 28.7 kPa in our case, so the representative pressure range for fatigue purposes is 0.69 × 28.7 = 19.8 kPa (0.0198 MPa). The faces of the blade corrugations in figure 10 are 1.111 m wide, and they are 15 mm (0.015 m) thick, so the local bending stress range comes to

In ([10], fig. 2–8), the nominal fatigue life (including safety margin) for a stress range of 70 MPa is 100 million cycles, so the fatigue life under local bending stresses comes to (100/1.7) × (70/54)^{5} = 215 years. The blade corrugations also act as beams spanning between the bulkheads in the wheel; the beam bending stress range comes to

The maximum bending stress is at the ends of the beam, at the weld to the bulkhead. The stress range for a fatigue life of 10^{8} cycles is given in ([10], fig. 2–8) as 30 MPa (class E welded joint), but this could be improved by grinding the weld toes to give a comparable fatigue life to that under local bending.

From the wheel blades the load path is through the bulkheads and then along the hub, which acts in torsion, to the end bulkheads (which are corrugated because they are a pressure barrier). These are in-plane loads, producing relatively small stresses. It is the end of the load path, at the calipers, which is a stress concentration. The torque on the wheel at a pressure difference of 19.8 kPa is 0.0198 × 10 × 50 × 10 = 99 MN m, so the force on each caliper is (99/2)/15 = 3.3 MN. This is carried over approximately 5 m of disc 20 mm thick, so the stress is approximately 3.3/(5 × 0.02) = 33 MPa, which is acceptable from the fatigue point of view.

- Received September 12, 2017.
- Accepted December 15, 2017.

- © 2018 The Author(s)

Published by the Royal Society. All rights reserved.