Page 83 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

Tidal Energy Chapter | 3 75

+
u dis , p dis
u w , p o
u o , p o
u h
u t

−
u dis , p dis

Upstream, S1 Turbine (disk), S2 Wake, S3
FIG. 3.27 Stream tube around a horizontal axis tidal turbine. The cross-sections for implementa-
tion of the momentum and energy equations have been also shown (S1, S2, and S3).

unknown, we can apply the energy equation in the upstream (between S1 and
S2) or the downstream part (between S2 and S3) of the disk.
Applying the energy equation between S1 and S2, leads to
u 2 o p o u 2 dis p + p o u 2 dis p + u 2 o
dis
dis
+ = + ⇒ = + − (3.37)
2g ρg 2g ρg ρg 2g ρg 2g
Similarly, for the downstream sections (S2 and S3), we have
u 2 p − u 2 p o p o u 2 p − u 2
dis dis w dis dis w
+ = + ⇒ = + − (3.38)
2g ρg 2g ρg ρg 2g ρg 2g
therefore
u 2 p + u 2 u 2 p − u 2
dis dis o dis dis w
+ − = + − (3.39)
2g ρg 2g 2g ρg 2g
which leads to
1 2 2
p + − p − = ρ(u − u ) (3.40)
dis dis o w
2
By equating the RHS of Eqs (3.36), (3.40), we can write
1 2 2
ρ(u − u ) = ρu dis (u o − u w ) (3.41)
o
w
2
or
1
(u o + u w ) = u dis = u o (1 − a) (3.42)
2