Prove bernaulis principle ?????
Answers
Explanation:
Bernoulli's Theorem is based on the conservation of mechanical energy of the fluid flows through the circular pipe and it states that Potential energy, kinetic energy and pressure energy of the fluid is conserved at two given point during the flow of fluid.
The following diagram shows the flow of the pipe from inlet to outlet through the circular pipe.
Flow of fluid
Fluid flows a distance
Δ
l
1
from inlet (as shown in above figure)
Fluid flows a distance
Δ
l
2
from an outlet (as shown in the above figure)
Write the expression for the work done by gravity.
W
g
r
a
v
i
t
y
=
−
m
g
(
h
2
−
h
1
)
Write the expression for the work done on fluid at the outlet point.
W
o
u
t
l
e
t
=
F
×
d
Here,d is displacement and f stands for force.
Substitute the known value in above equation.
W
o
u
t
l
e
t
=
−
(
A
2
×
P
2
×
Δ
l
2
)
Here
A
2
is area of outlet and
P
2
is pressure at outlet.
Write the expression for the work done on fluid at inlet point.
W
i
n
l
e
t
=
F
×
d
Here F is force and d is displacement.
Substitute the known value in above equation.
W
i
n
l
e
t
=
A
1
×
P
1
×
Δ
l
1
Here
A
1
is area of inlet and
P
1
is pressure at inlet.
And by work energy principle net work is calculated as,
W
n
e
t
=
Δ
K
.
E
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(
I
)
Write the expression for the net work done.
W
n
e
t
=
W
i
n
l
e
t
+
W
o
u
t
l
e
t
+
W
g
r
a
v
i
t
y
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(
I
I
)
By comparing the equation (I) and (II)
W
i
n
l
e
t
+
W
o
u
t
l
e
t
+
W
g
r
a
v
i
t
y
=
Δ
K
.
E
Substitute the known value.
A
1
×
P
1
×
Δ
l
1
−
(
A
2
×
P
2
×
Δ
l
2
)
−
m
g
(
h
2
−
h
1
)
=
1
2
m
(
v
2
2
−
v
1
2
)
Substitute the
ρ
×
A
1
×
Δ
l
1
or
ρ
×
A
2
×
Δ
l
2
for m both values are equal.
A
1
×
P
1
×
Δ
l
1
−
(
A
2
×
P
2
×
Δ
l
2
)
−
ρ
×
A
1
×
Δ
l
1
g
(
h
2
−
h
1
)
=
1
2
ρ
×
A
1
×
Δ
l
1
(
v
2
2
−
v
1
2
)
Hence, the Bernoulli's equation is
P
1
−
P
2
−
ρ
g
(
h
2
−
h
1
)
=
1
2
ρ
(
v
2
2
−
v
1
2
)
By rearranging the above equation bernoulli's equation written as,
P
1
+
1
2
ρ
v
1
2
+
ρ
g
h
1
=
P
2
+
1
2
ρ
v
2
2
+
ρ
g
h
2
Thus the Bernoulli's equation is
P
1
+
1
2
ρ
v
1
2
+
ρ
g
h
1
=
P
2
+
1
2
ρ
v
2
2
+
ρ
g
h
2