Page 53 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

44 Fundamentals of Ocean Renewable Energy

FIG. 2.3 Schematic of a streamtube control volume for the derivation of Bernoulli’s equation.

− u in A in + u out A out = 0 ⇒ Q = uA = cons. (2.78)
where ‘in’ and ‘out’ subscripts denote the inflow and outflow, respectively. Q is
the volumetric flow rate or discharge, which is constant for this case. Note that
the inner product of the velocity and the surface vector for inflow is negative,
whilst for outflow it is positive.
For the 1D momentum equation, in the steady case, we have
d(mu)
F = = ρuu · dS ⇒ u(u · A) = ρuQ (2.79)
dt S
where Q is positive for outflux and negative for the influx of momentum. The
momentum equation can be simply written as

F = ρQ(u out − u in ) (2.80)
Another useful equation is Bernolli’s equation, which is the energy equation for
inviscid 1D flows in the steady-state case. Bernoulli’s equation can be easily
derived from the momentum equation. Fig. 2.3 shows an infinitesimal control
volume of a streamtube. The forces acting on this control volume include weight
and pressure. From calculus, we can write this relation for any function for small
values of δs
∂f
f(s + δs) ≈ f(s) + δs (2.81)
∂s
Therefore, the summation of forces acting on the control volume can be
formulated as

F =−ρgdV cos θ + pA| − pA| s+ds
s
∂(pA)

=−ρg cos θAds + pA + pA + ds (2.82)
∂s
where ds = δs and A are the length and the area of the control volume,
respectively. θ represents the angle of the streamtube to the vertical direction, as