Draw the schematic diagram of the variable area horizontal pipe.
Here, the direction of the stream line is s.
Step 2/6
Write the equation for the velocity of the centerline as follows:
…… (1)
Here, the unit vector
indicates that the velocity is in x direction.
Write the velocity of flow of water along the x direction.
Differentiate the above equation partially with respect to x.
Step 3/6
Obtain the density and the specific weight of water from table, “Approximate Physical
Properties of Some Common Liquids (BG Units)” in the Appendix in the text book.
Density,
Specific weight,
Write the equation of motion along the stream line direction as follows:
…… (2)
Here, the specific weight of water is
, the inclination of the flow is
gradient along the x direction is
, the density of the water is
gradient along the x direction is
.
Step 4/6
Substitute
,
for
for V and
,
for
for
,
for
in equation (2).
Therefore, the pressure gradient needed to produce this flow is
(b)
Write the pressure gradient along the flow.
, the pressure
, and the velocity
Integrate equation between two points 1 and 2 as follows:
Substitute
for
.
Therefore, the pressure at section (2) is
Step 5/6
(ii)
Calculate magnitude of velocity at section (1).
Substitute
for x.
.
Calculate magnitude of velocity at section (2).
Substitute
for x.
Step 6/6
Apply Bernoulli’s equation between sections (1) and (2) as follows:
Here, the pressures at sections (1) and (2) are
points (1) and (2) are
are
and
and
and
respectively, the velocities at
, the elevation heads of sections (1) and (2)
respectively .
Here, the pipe is horizontal and hence, the elevation heads of sections (1) and (2) is the
same, which means
Substitute
for
.
,
for
,
for
, and
for
,
Therefore, the pressure at section (2) using Bernoulli’s equation is
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90
2
Draw the velocity profile and the direction of velocity as follows:
Step 2/7
Consider that the velocity profile is uniform.
Write the equation of motion normal to the streamline direction as follows:
.
Here,
is the flow in normal direction, is the specific weight of the water,
is the
density of the water, R is the local radius of curvature, and
is the velocity of water.
Substitute
for
(When the analysis is normal to the stream line,
at the very bottom of the bend, dz and dn are collinear).
because
The radius of curvature for circular streamline is 6m. So, the radius of curvature for any
point at a distance n from the base is
Substitute
.
for R.
…… (1)
Step 3/7
Integrate equation (1), with limits of the pressure ranging from
of n ranges from 0 to n.
to
and the value
…… (2)
Here, the pressure at point (1) is
.
Step 4/7
Apply boundary conditions as follows:
When
, then pressure
Thus equation (2), becomes as follows:
Substitute 1 m for
and
for
in equation (2).
…… (3)
Step 5/7
Calculate the pressure at point (2) as follows:
Substitute
for ,
and
for n in the equation (3).
for g,
for
,
for
,
Therefore, the pressure at point (2) is
.
Step 6/7
Apply boundary conditions as follows:
When,
, then pressure,
.
Thus equation (2), becomes as follows:
Substitute 2 m for
and
for
in equation (2).
…… (4)
Step 7/7
Calculate the pressure at point (3) as follows:
Substitute
and
for
for ,
for g,
in the equation (4).
for
,
Therefore, the pressure at point (3) is
.
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156
4
Consider free surface and exit condition of water surface and apply Bernoulli’s relation.
Here, the pressure acting at free surface inside the tank is
, the density of the water
in the tank is , the acceleration due to gravity is , the velocity of the flow at free
surface is
, the height of free surface from bottom of the tank is
exit of the tank is
, the velocity of flow at exit of the tank is
the tank from bottom surface is
, the pressure at
, and the height of exit of
.
Step 2/2
Note that the pressure is the same at inside and outside of the tank.
Substitute
for
,
for
, 0 for
,
for
, and
for g.
Therefore, the height of free surface form exit of tank is
Draw the diagram of pipe to show the considered points (1) and (2).
Step 2/6
Apply continuity equation to the system as follows:
Here,
is the area of the pipe at section (1), section (2) and
flow at section (1) and section (2) respectively.
Substitute
for
Substitute 0.1 m for
Step 3/6
and
.
for
.
is the velocity of
Write the relation for pressure at point (1).
Write the relation for pressure at point (2).
Use the Bernoulli’s equation between point 1 and point 2.
Here,
is the pressure, velocity, datum at point1 and
velocity, datum at point2 respectively.
Since, point (1) and (2) are at the same elevation:
...... (1)
Step 4/6
Substitute
for
,
for
, and
for
in equation (1).
is the pressure,
Step 5/6
Substitute
for
.
Step 6/6
Calculate the flow rate through the pipe.
.
Substitute
for
, 0.1 m for
Therefore, flow rate through the pipe is
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33
35
Draw the schematic diagram of the tank.
, and
for g.
.
Step 2/9
Calculate the area of the hose pipe by using the following relation:
Here,
is the diameter of the hose pipe.
Substitute
for
.
Calculate the area of the nozzle at point
Here,
by using the following relation:
is the diameter of the nozzle.
Substitute
for
.
Step 3/9
Apply the Bernoulli’s equation at points
and
.
Here,
are the pressures at points
and
,
is the specific weight of
are the velocities at points
and
,
is the acceleration due to
water,
,
,
gravity, and
Substitute
and
for
is the difference in heads of points
for
,
.
for
and
, (since atmospheric pressure), 0 for
.
,
for
,
Step 4/9
Calculate the flow rate from the tank by using the following relation:
Substitute
for
and
for
.
Therefore, the flowrate from the tank is
.
Step 5/9
Apply the Bernoulli’s equation at points
Here,
is the pressure at point
velocity at point
Substitute
and
for
for
.
is the datum head at point
, and
.
, 0 for
,
for
,
for
is
.
.
Therefore, the pressure at point
Step 6/9
,
and
,
for
,
is the
Using the equation of continuity, the following relation holds good.
Here,
is the velocity at point
Substitute
for
.
,
for
, and
for
.
Step 7/9
Apply the Bernoulli’s equation at points
Here,
is the pressure at point
velocity at point
Substitute
and
for
for
,
and
.
is the datum head at point
, and
is the
.
, 0 for
,
for
,
for
,
.
Therefore, the pressure at point
is
.
Step 8/9
Using the equation of continuity, the following relation holds good.
for
,
Here,
is the velocity at point
Substitute
for
.
,
for
, and
for
.
Step 9/9
Apply the Bernoulli’s equation at points
Here,
is the pressure at point
velocity at point
Substitute
and
for
for
,
and
.
is the datum head at point
, and
is the
.
, 0 for
,
for
,
for
.
Therefore, the pressure at point
is
.
,
for
,