# The Physics of Energy sources Renewable sources of energy. Tidal power

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1 The Physics of Energy sources Renewable sources of energy Tidal power B. Maffei Renewable sources III 1

2 Tide principle! Essentially due to the Moon-Earth system The Moon-Earth system revolves around their centre of gravity (O on the figure) If we have:! The various distances L, L, r and D has shown on the figure! M mass of the Moon! M mass of the Earth Kepler + Newton à Gravity force Inertia force GMM ʹ D ML M ʹ L ʹ Lʹ MLω M L ω MD ( M ʹ + M ) ʹ ʹ 467km M7.35x1 kg M5.97x1 4 kg r6.37x1 6 m D3.84x1 8 m So the centre of gravity of the system is located within the Earth radius Renewable sources III

3 Tide principle ()! We cannot assume that the Earth s mass is concentrated at its centre E only! Not all the elements of mass are at the equilibrium position! At position X (closer to the Moon) increase of gravity force and decrease of inertia! At position Y (further from the Moon) decrease of gravity force and increase of inertia Consider an element of mass Δm on different points of the Earth GMΔm ΔmLʹ ω D F F Δm Lʹ + r ω F Y X Y F X ( ) GMΔm ( D r) At E: equilibrium GMΔm ( D + r) ( Lʹ ) + Δm r ω At Y: resulting force F Y directed away from Moon At X: resulting force F X directed towards Moon F X F Y We get: F X F Y Δmrω 1+ Lʹ D (To be shown as an exercise) forces of equal amplitude and opposite directions Renewable sources III 3

4 Resulting effect! Deformation will be limited for solids! The impact will be greater on liquids! It will lead to oceans deformation! Taking into account the Earth s rotation! We should expect lunar tidal ranges each day of equal amplitude! However this is a simplified model! Many effects will impact the amplitude and period of the tides! Earth rotation axis is not perpendicular to the ecliptic plane! The tides cannot move fast enough to keep up with the Earth s rotation Lags behind and is not in phase anymore! Moon-Earth distance varies across the year! The Sun also induces a tidal effect Even if smaller (about 1/3) it is out of phase and creates perturbations The amplitude is not constant (but predictable). Period is 4h5min Renewable sources III 4

5 Power production! The change in height between successive high and low tide! Resulting range R! It is around.5m 1m at sea not that useful! Can be up to 1m high in particular locations Near continental land masses In this case we can make use of tidal range power (potential energy)! Movement of water produces tidal currents! By funneling, these currents can be increased! They can have useful speeds close to coasts and inter-island channels! We can obtain tidal current/stream power (kinetic energy). Renewable sources III 5

6 Tidal current power! The power of tidal currents may be harnessed in the same way than wind power Similarly to wind power, the electrical power density from water current is: q C p! Funneling creates strong current where a turbine can be located ρ u 3 ρ :the water density Much higher than wind Needs smaller turbine radius u: current speed C p : efficiency- in theory no more than 6% - 4% in practice Tidal current speed varies with time as: u u sin π t τ τ : period between max 1h 5mins u : max current speed Assuming C p.4 Average electrical power per unit of cross section q C p ρ u t τ / 4 3 π t sin 3 dt τ t 3 τ 4 3.ρ u / 4.1ρ u t τ t dt 3π τ Ex: assuming a max current of 3m/s with a turbine of 5m radius A78.5m -1 3 ( 3m.s ) kw 3-3 A q A.1ρ u 78.5m.1 15kg.m 17 In some locations max speed of 5m/s leading to 1MW of power Renewable sources III 6

7 Tidal range power! The goal here is to trap water from high tide in a basin! Water will then run through a turbine at low tide Similar process than hydroelectricity If the basin has a surface area A and the tide a range R, the mass of water trapped in basin is ρar At low tide this mass of water will drop by a height R/ The maximum potential energy available per tide is then: P ρ. A. g τ ( R + ) max Rmin Mean High tide level Low tide level Averaged over the tidal period τ, we get the average potential power ρ. A. g R τ energy per tide ( ρ. A. R) g Renewable sources III 7 R P ρ. A. R g τ However, the tide range is not constant over the month. This variation is sinusoidal. If we have a mean range R Mean and R varying between R max and R min then: Ex: With a basin of 1 square km and a range min/max of.5m/5m, P1.6MW Centre of gravity at R/ R

8 Conclusion on tidal power! Tidal range power has been used for a while! Small scale for centuries: medieval Europe, China! First large scale power plant in 1965 La Rance St Malo: 4MW of electric power! About 1GW of electrical power is produced in the World! Tidal stream power use relatively new technology! A few devices have been developped! Prototypes have shown high efficiency (up to 6% of tidal energy converted)! Prototypes producing between 3kW up to 3.5MW (Australia)! Advantages! Unlike solar or wind power, tidal power production is predictable.! Studies are showing that it is costly competitive.! Totally renewable and clean! Disadvantages! Location dependant cannot be anywhere! Large capital costs! Potential ecological harm! Variable power production during the day Renewable sources III 8

9 References! Most of the material of this lecture is coming from! Ref4: Renewable energy resources, J. Twidell and T. Weir, second edition, 6 Renewable sources III 9

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