Page 43 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

34 Fundamentals of Ocean Renewable Energy

FIG. 2.1 Schematic of a control volume.

with time. However, if the inflow and outflow of water are not equal, they lead
to a change in the amount of water in that pool/lagoon. For this simple case, we
can write
ρdV pond
ρQ out − ρQ in + = 0 (2.13)
dt
Note that since the mass of a system does not change, dN = dm = 0inthe
dt dt
previous equation. The Reynolds transport theorem can be expressed in index
notation as follows
dN d ∂(ρβ)
= (ρβ)dV = dV + ρβu i n i dS (2.14)
dt dt V ∂t S
V system
To change the integral form of an equation to a differential form, we need to use
a theorem which relates the surface integral to the volume integral. Referring
to vector algebra, using Gauss’s theorem (or Divergence theorem), the integral
over the control surface can be replaced with the integral over the volume as
follows

u · dS = (∇· u ) dV (2.15)
S V
∂u i
where ∇· u = is the divergent operator. In indicial notation, using Gauss’s
∂x i
theorem, we can write
∂(ρβu i )

ρβu i n i dS = dV (2.16)
S V ∂x i
Thus, the Reynolds transport theorem may also be written as
dN ∂(ρβ) ∂(ρβu i )
= + dV (2.17)
dt V ∂t ∂x i