Environmental Engineering Reference
In-Depth Information
the sum of the potential and pressure energy in the
z
-direction is:
1
2
ρv
2
ρgz
+
p
=
constant. The kinetic energy of a particle is equal to
The total energy for an entire cross section is equal to the sum of the
energy of all the fluid particles. To find the total kinetic energy in this
particular cross section, the velocity is expressed by the mean velocity:
v
mean
=
Q/A
. However, in open channels the velocity is not uniformly
distributed over the depth and the width due to the presence of the free
water surface and the friction along the boundary (bottom and sides).
For the total energy head this fact is taken into account by multiplying
the velocity head (
v
2
/2
g
) by the coefficient
α
, the Coriolis coefficient
(see Section 2.5). The true average kinetic energy across the cross-section
per unit of volume is: (
2
ρv
2
)
average
=
v
2
.
The total energy
E
passing through a cross section is:
1
2
αρ
¯
1
2
αρ
¯
v
2
E
=
+
ρgz
+
p
(2.15)
The total energy
E
divided by the weight results in the energy head,
being the total energy per unit weight.
v
2
2
g
+
¯
α
p
ρg
+
E
tot
=
z
(2.16)
If you assume two cross sections perpendicular to straight and parallel
streamlines, and that the energy loss is negligible (the energy principle of
Bernoulli), then:
α
1
v
1
α
2
v
2
p
1
ρg
+
p
2
ρg
+
E
tot
=
2
g
+
z
1
=
2
g
+
z
2
(2.17)
where:
z
=
height of the channel bottom in m above datum
p/ρg
=
pressure head in a cross-section in m
αv
2
/
2
g
=
velocity head in a cross-section (kinetic energy) in m
total energy head in a cross-section in m above a datum
In a
real fluid
there are always friction and/or local losses (
E
E
tot
=
=
h
f
)
v
1
v
2
α
1
¯
p
1
ρg
+
α
2
¯
p
2
ρg
+
2
g
+
z
1
=
2
g
+
z
2
+
h
f
(2.18)
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