Page 88 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

80 Fundamentals of Ocean Renewable Energy

v 2
= h L (t) − h O (t) ⇒ v = 2g [h L (t) − h O (t)] (3.51)
2g
where v is the water velocity at the turbine section. Therefore, the flow rate or
discharge through the tidal turbines will be

Q t = av = a 2g [h L (t) − h O (t)] (3.52)
where a is the swept area of all turbines. As water flows through the turbines
to the ocean, the water elevation inside the tidal lagoon gradually drops. Using
the continuity equation, we can find a relationship between the change in water
elevation inside the lagoon and the outflow rate. Consider the volume of water
which leaves the lagoon during δt
A L dh L
–
δ V = Q(t)δt =−A L δh L ⇒ Q =− (3.53)
dt
Now, we can derive an equation to estimate the theoretical power through
the turbines. For a turbine, we know that the power at any time t will be
1 3 1 2
P = ρav = ρQv (3.54)
2 2
Referring to Eq. (3.51) and replacing v, the power can also be evaluated by
water elevations as follows
P = ρQ(t)g [h L (t) − h O (t)] (3.55)
replacing Q (Eq. 3.53), leads to

dh L
P =−ρA L g [h L (t) − h O (t)] (3.56)
dt
Eq. (3.56) indicates that the theoretical power at each point in time is
proportional to the difference of water elevation inside and outside a lagoon,
and the rate of change of water surface elevation inside the lagoon (or flow
discharge). Finally, if we have the time series of water elevation inside and
outside a tidal lagoon (i.e. ocean), the total theoretical energy, which passes
through turbines from t 1 to t 2 , can be calculated as

t 2
dh L
E = −ρA L g [h L (t) − h O (t)] dt (3.57)
dt
t 1
Consider a special case where the discharge and the surface area of a lagoon
remain constant during a period (i.e. Q =−A L dh L /dt = const), then
t 2

E = ρgQ [h L (t) − h O (t)] dt (3.58)
t 1