Page 84 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

76 Fundamentals of Ocean Renewable Energy

In other words, the axial velocity at the disk is simply the average of the
velocities at the upstream and downstream sections. Eq. (3.42) also leads to

u w = (1 − 2a)u o (3.43)

3.13.1 Power Coefficient and the Betz Limit
Now that we have computed the forces and velocity at the actuator disk, we can
evaluate the extracted power using Eq. (3.36) as follows
−
+
P = Fu dis =[p − p ]A dis u dis = A dis ρ[(u o − u w )]u 2 (3.44)
d d dis
Referring to Eqs (3.42), (3.43) and replacing u w and u dis , leads to
2
3
P = ρA dis (u o − [(1 − 2a)u o ])[u o (1 − a)] = 2ρA dis u a(1 − a) 2 (3.45)
o
Considering the flux of kinetic energy through the swept area of the turbine,
the total available power (before the turbine influences the flow field) is P ava =
1 3
o
2 ρA dis u , whilst the extracted power is
3

3 2 1 3 2 2
P = ρA dis u a(1 − a) = ρA dis u o [4a(1 − a) ]=[4a(1 − a) ]P ava
o
2
(3.46)
The power coefficient, C p , may be defined as the ratio of extracted power to
available power, and is given by
P 2
C p = = 4a(1 − a) (3.47)
P ava
Fig. 3.28 plots the power coefficient as a function of axial flow induction
factor. As this figure shows, the maximum power coefficient is 0.593, corre-
sponding to a = 1/3. Alternatively we can write
dC p 2 1
= 4(1 − a) − 8a(1 − a) = 0 ⇒ a = (3.48)
dp 3
16
which leads to C p = = 0.593. This limit is called the Betz limit after the
27
German physicist Albert Betz. Therefore, theoretically, no turbine can convert
more than 59.3% of the kinetic energy in a tidal current into mechanical energy
to turn a rotor.

3.14 TIDAL RANGE: LAGOONS AND BARRAGES

In contrast to tidal stream technology (Section 3.12), the extraction of tidal
energy using tidal lagoons (or tidal barrages) is based on the simple con-
cept of converting potential energy to kinetic energy. This concept is quite
old and has been largely implemented in hydropower projects for centuries.