Basics of Renewable Energy Supply - Renewable Energy: Technology, Economics and Environment - page 71

Power and work capacity of water. Due to gravitation water flows within a
stream or a river from a higher geodesic site to a lower geodetic site. At both sites
the water is characterised by a particular potential and kinetic energy which is
different from each other. In order to identify this energy difference of the out-
flowing water, in an approximation, a stationary and friction-free flow with in-
compressibility can be assumed. With these preconditions the hydrodynamic Ber-
noulli pressure equation can be applied, and written according to Equation (2.22).

1

2

p + ρ g h + ρ v = const . (2.22)

2

Wa Wa Wa

p is the hydrostatic pressure, ρ Wa the water density, g the acceleration of grav-
ity, h the head and v Wa the velocity of the water flow. Equation (2.22) can be con-
verted in a way that the first term expresses the pressure level, the second term the
site level and the third term the velocity level (Equation (2.23)).

2

p 1 v

+ h +

Wa

= const .

(2.23)

ρ g 2 g

Wa

This Equation (2.23) for example, allows determining the utilisable head h util of
a particular section of a stream or river. It is calculated according to Equation
(2.24) from the pressure differences, the geodesic difference in height and the
different flow velocities of the water. When applying this formula we need to bear
in mind that this is an idealised form of analysis that does not consider any actual
losses. Under actual conditions therefore, from the utilisable head, also the head
losses resulting from friction of the individual water molecules among each other
and the surrounding matter, have to be subtracted (see Chapter 7).

2 2

v − v

p − p

( )

Wa , Wa , 2

h =

1 2

+ h − h +

(2.24)

util 1 2

ρ g 2 g

Wa

As the terms defined by the differences in pressure and in velocity are usually
relatively small, the geodesic head between the two water surfaces in a water
course (e.g. stream, river) can generally be used as the utilisable head in a first
rough estimation. The other elements of Equation (2.24) mainly occur within the
hydraulic system of a hydro power station.
Starting from this assumption, the power P Wa resulting from the respective wa-
ter supply can be calculated using Equation (2.25). q Wa is the volume-related flow
rate. According to this formula the product of flow and utilisable head basically
determine the power of the water. Large heads can generally be achieved in
mountainous areas, whereas in lowland areas mainly the flow assumes high val-
ues.