Ya cannae get aroond conservation o'
energy, ya reekie oxtered sassenachs yae. (Translation from the
Scots: You can't get around conservation of energy).
Conservation of energy from a far
field perspective
Consider an arbitrary boundary around a
big bit of sea. In order for a wave energy converter (WEC) to absorb
energy from within that boundary, the waves leaving the boundary must
have less energy than the waves entering the boundary. The only
mechanism to capture useful energy while ensuring less energy in the
waves leaving the boundary, is to create waves that destructively
interfere with waves that would leave the system in the absence of
energy capture. To create waves while absorbing energy involves the
radiation of waves by damped (by a power take off system that is)
wave excited motion.
Conservation of energy from the
device perspective
Consider the boundary between a WEC and
the sea. In order for the WEC to absorb energy from within that
boundary, the energy leaving must be less than that entering. The
only mechanism that ensures that the energy entering the system (wave
excited oscillation) is less than the energy leaving the system
(oscillation induced radiation), is to damp this oscillation (with a
power take off system). Therefore, for energy capture, the
oscillations and the resulting radiated waves must be less than when
no energy is captured.
The paradox being...
Herein lies the radiation paradox: from the conservation of energy perspective, radiation of waves is necessary for power capture; the bigger the gap between the energy entering and leaving the far field and device boundaries, the more energy is available for capture. Yet to create these destructive radiated waves, it is necessary to radiate waves containing less power than when energy is not being captured. Although we can say 'it is necessary to radiate waves to capture waves', an oscillating body will radiate waves even when it is not capturing any energy. So it is equally valid to say 'it is necessary to radiate fewer waves to capture waves' (radiated waves need to be the right kind of course!)
Herein lies the radiation paradox: from the conservation of energy perspective, radiation of waves is necessary for power capture; the bigger the gap between the energy entering and leaving the far field and device boundaries, the more energy is available for capture. Yet to create these destructive radiated waves, it is necessary to radiate waves containing less power than when energy is not being captured. Although we can say 'it is necessary to radiate waves to capture waves', an oscillating body will radiate waves even when it is not capturing any energy. So it is equally valid to say 'it is necessary to radiate fewer waves to capture waves' (radiated waves need to be the right kind of course!)
Making sense of the paradox
The only way to make sense of this is
to consider Falne's standard formulation for power capture: absorbed
power is the difference between excited power and radiated power:
Pa = Pe- Pr.
Pe is proportional to oscillation (velocity, for a primary wave activated body with an oscillating position), and Pr is proportional to the square of the oscillation. Pa plotted against oscillation gives a parabola, with zero power at oscillations that are too large or small. This means that radiation of waves too large or too small will lead to ineffective power capture.
Pa = Pe- Pr.
Pe is proportional to oscillation (velocity, for a primary wave activated body with an oscillating position), and Pr is proportional to the square of the oscillation. Pa plotted against oscillation gives a parabola, with zero power at oscillations that are too large or small. This means that radiation of waves too large or too small will lead to ineffective power capture.
Control over the size of radiated wave
is exercised through choice of the amount of damping applied.
Unsurprisingly, the most effective value lies between the extremes of
zero damping (Pe= Pr = excitation and radiation
power for undamped response), and the damping required to reduce the
oscillations to zero (Pe= Pr= 0).
Image credits:
Rubber ducky X-ray copyright of Andrew
Liepzig http://planetzig.com/art.html
Hi Ally,
ReplyDeleteJust wanted to add another point. The Radiated waves also need to be in the same useful direction. An Axis-symmetric body will radiate waves in all direction, hence depending on the comparisons, should account for some dissipated energy if not more, even when resonating in harmony with the incident waves.
hi Vernash
ReplyDeleteIndeed the radiated waves do need to be in the right direction. Directionality is an important part of what makes a radiated wave 'the right kind'. Fig 6 of http://www.wavepowerconundrums.com/2013/01/dr-dr-i-see-waves-before-my-eyes.html shows a simplified 2D illustration of what happens when the waves are in the wrong direction - the ideas apply to the 3D case.