Environmental Engineering Reference
In-Depth Information
The resultant of the external forces is equal to the rate of change of
momentum of that body.
P
and
v
represent vectors; hence the change
in momentum has the same direction as the resultant. Equation 2.12 states
that the forces acting on a fluid mass are equal to the rate of change of the
momentum of that mass. The first term on the right side represents the net
flow rate of momentum going out of the control volume and the second
term represents the rate of accumulation of momentum within the control
volume during the time interval
t
.
The direction of
P
is the same as that of
v
;
P
represents the vec-
torial summation of all forces on the mass, including the gravity forces,
shear forces, and pressure forces including those exerted by fluid sur-
rounding the mass as well as the pressure forces exerted by boundaries in
contact with the mass. The equation always applies.
In the case of steady flow, the last term in equation 2.12 is equal to
zero, the force is equal to the net momentum outflow across the control
surface and the equation becomes:
P
d(
m
v
)
out
−
d(
m
v
)
in
d(
m
v
)
out
d
t
d(
m
v
)
in
d
t
=
=
−
(2.13)
d
t
P
=
ρQ
(
β
2
v
2
−
β
1
v
1
)
(2.14)
where:
P
=
sum of all external forces acting on the control body (
N
)
discharge (m
3
/s)
Q
=
area of the cross section (m
2
)
A
=
v
=
mean velocity perpendicular to the cross-section (m/s)
m
=
mass of water passing the cross section (kg)
m
=
ρvA
β
=
Boussinesq coefficient (see Section 2.5)
In this equation, the factor
β
is the coefficient of Boussinesq, which
incorporates the effect of the velocity distribution on the average veloc-
ity. When there are no external friction forces, only the conditions at
the end sections of the control volume govern the impulse-momentum
principle.
Energy principle
The energy equation holds true as long as proper allowance is made for the
energy losses. The total energy of a fluid particle per unit of volume is
the sum of three types of energy (kinetic, potential and pressure). For an
open channel with steady flow, and with straight and parallel streamlines,
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