89
2.5 Exercises
Study the energy exchange. An expression for the energy increase in the driven
flow has already been formulated. Determine the mechanical energy decrease in the
driving jet flow 01
03
( p
p ) / r
−
. In most applications, the driving water for the jet
is tapped from the flow in the pressure pipe. Driving water is then produced with a
pump increasing the total pressure p 03 to the total pressure p 01. With an ideal pump,
the energy increase required exactly equals the mechanical energy decrease within
the driving jet between the injection point (1) and the jet pump outlet (3). The ratio
of the energy increase in the driven flow to the energy decrease in the driving flow
may then be defined as the efficiency η of the jet pump. Note that this efficiency
definition also applies with driving water from another origin.
Determine the efficiency η for varying α and U. Study the values of a = 0.2, 0.4,
0.6, 0.8 combined with the values of U = 0, 0.25, 0.50, 0.75, 1. Observe that the efficiency is weakly dependent on a, but strongly dependent on U. To attain a good
efficiency (better than 0.75), U must exceed about 0.6. This implies that the velocity
difference between driving and driven flow at the mixing position must not be too
big. This is obvious, as the loss considered is mixing loss. The driven jet must thus
be highly accelerated at the position of the injector. See the geometry of a real jet
pump as shown in the figure below. The velocity of the driven jet must mostly be
limited in order to avoid cavitation. For instance, to v 2 = 10 m/s corresponds a pressure decrease in the suction pipe of
2
2
v / 2 50000 Pa 0.5 bar
r
=
=
. Such a strong
pressure decrease implies that the geometrical suction height must be under 5 m.
Jet pumps are therefore mostly mounted below the water surface in the suction well
(submerged). A pressure drop of 0.5 bar is then no problem. A diffuser beyond the
mixing chamber is required as well. In the example quoted, velocity exceeds 10 m/s
after mixing, but the velocity in a pipe is typically at maximum about 2 m/s. So
the diffuser generates substantial losses. Attaining a considerable pressure increase
with good efficiency is thus impossible.
For the study of pressure increase and mass flow rate in the driven flow,
we define still other parameters. As the measure for the pressure increase, we
define the ratio δ of the total pressure increase in the driven flow to the total
pressure decrease in the driving flow. As the measure for the mass flow rate, we
define the ratio μ of the mass flow rate of the driven flow to the mass flow rate
of the driving flow. With these definitions, the efficiency is η = μδ. Determine δ
and μ for varying α and U. Study again the values of a = 0.2, 0.4, 0.6, 0.8 combined with the values of U = 0, 0.25, 0.50, 0.75, 1. Observe that, for a similar
efficiency, as well a low mass flow rate together with a high pressure increase as
a high mass flow rate together with a low pressure increase may be chosen. The
choice of the combination is mainly determined by the parameter α with only a
weak effect of the value of U. Since the analysis takes neither the friction loss in
the mixing chamber nor the diffuser loss into account, the velocity ratio should
be chosen lower in practice than the values obtained from the analysis. Then,
velocities in the mixing chamber and in the diffuser decrease. A practical value
of U is about 0.5. The efficiency obtained in the analysis is about 0.65, arriving
at about 0.35 in practice.
2.5 Exercises
Study the energy exchange. An expression for the energy increase in the driven
flow has already been formulated. Determine the mechanical energy decrease in the
driving jet flow 01
03
( p
p ) / r
−
. In most applications, the driving water for the jet
is tapped from the flow in the pressure pipe. Driving water is then produced with a
pump increasing the total pressure p 03 to the total pressure p 01. With an ideal pump,
the energy increase required exactly equals the mechanical energy decrease within
the driving jet between the injection point (1) and the jet pump outlet (3). The ratio
of the energy increase in the driven flow to the energy decrease in the driving flow
may then be defined as the efficiency η of the jet pump. Note that this efficiency
definition also applies with driving water from another origin.
Determine the efficiency η for varying α and U. Study the values of a = 0.2, 0.4,
0.6, 0.8 combined with the values of U = 0, 0.25, 0.50, 0.75, 1. Observe that the efficiency is weakly dependent on a, but strongly dependent on U. To attain a good
efficiency (better than 0.75), U must exceed about 0.6. This implies that the velocity
difference between driving and driven flow at the mixing position must not be too
big. This is obvious, as the loss considered is mixing loss. The driven jet must thus
be highly accelerated at the position of the injector. See the geometry of a real jet
pump as shown in the figure below. The velocity of the driven jet must mostly be
limited in order to avoid cavitation. For instance, to v 2 = 10 m/s corresponds a pressure decrease in the suction pipe of
2
2
v / 2 50000 Pa 0.5 bar
r
=
=
. Such a strong
pressure decrease implies that the geometrical suction height must be under 5 m.
Jet pumps are therefore mostly mounted below the water surface in the suction well
(submerged). A pressure drop of 0.5 bar is then no problem. A diffuser beyond the
mixing chamber is required as well. In the example quoted, velocity exceeds 10 m/s
after mixing, but the velocity in a pipe is typically at maximum about 2 m/s. So
the diffuser generates substantial losses. Attaining a considerable pressure increase
with good efficiency is thus impossible.
For the study of pressure increase and mass flow rate in the driven flow,
we define still other parameters. As the measure for the pressure increase, we
define the ratio δ of the total pressure increase in the driven flow to the total
pressure decrease in the driving flow. As the measure for the mass flow rate, we
define the ratio μ of the mass flow rate of the driven flow to the mass flow rate
of the driving flow. With these definitions, the efficiency is η = μδ. Determine δ
and μ for varying α and U. Study again the values of a = 0.2, 0.4, 0.6, 0.8 combined with the values of U = 0, 0.25, 0.50, 0.75, 1. Observe that, for a similar
efficiency, as well a low mass flow rate together with a high pressure increase as
a high mass flow rate together with a low pressure increase may be chosen. The
choice of the combination is mainly determined by the parameter α with only a
weak effect of the value of U. Since the analysis takes neither the friction loss in
the mixing chamber nor the diffuser loss into account, the velocity ratio should
be chosen lower in practice than the values obtained from the analysis. Then,
velocities in the mixing chamber and in the diffuser decrease. A practical value
of U is about 0.5. The efficiency obtained in the analysis is about 0.65, arriving
at about 0.35 in practice.