VVR 120 Fluid Mechanics
7. Basic equations II (4.7-4.8)
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•
•
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Bernoulli´s equation
Kinetic energy, potential energy, and pressure energy
for fluid in motion
Energy head (line): velocity head, pressure head, and
elevation head,
Bernoulli´s equation, applications
Exercises: C12, C14, and C23
VVR 120 Fluid Mechanics
BERNOULLI’S EQUATION
• Bernoulli’s equation is the energy equation for an ideal
fluid (friction and energy losses assumed negligible).
• Bernoulli’s equation may, however, be used with
satisfactory accuracy in many engineering problems and
has the advantage of providing valuable insight about
energy conditions in fluid flow
VVR 120 Fluid Mechanics
BERNOULLI’S EQUATION
p
γw
+
V
2
+ z = H = const .
2g
Bernoulli’s equation is a useful relationship
between pressure p, velocity V, and
geometric height z, above a reference
plane (datum).
H: energy head (m)
z: elevation head above datum (m)
V: velocity (m/s)
g: gravity acceleration (m/s2)
p: pressure (Pa)
w: weight density for the flowing fluid
(N/m3)
Quantity
Name
Measure of
H
Energy head
Total energy
P/w
Pressure head
Pressure energy
Z
Elevation head
Potential energy
V2/(2g)
Velocity head
Kinetic energy
p
γ
w
+ z = piezometric head or
H .G.L = Hydraulic Grade Line =
Trycknivå
VVR 120 Fluid Mechanics
p/w
VVR 120 Fluid Mechanics
Validity criterias – Bernoulli’s equation
1)
2)
3)
4)
Along a streamline
For an ideal fluid
Steady flow
Incompressible flow
VVR 120 Fluid Mechanics
C12 If crude oil flows through this pipeline
and its velocity at A is 2.4 m/s, where is the
oil level in the open tube C?
VVR 120 Fluid Mechanics
KINETIC ENERGY CORRECTION FACTOR, α
For a real fluid, friction will cause a non-uniform velocity distribution ⇒
the velocity head have to be corrected before use of the Bernoulli equation.
The real kinetic energy is obtained by integration over the section area and is
then expressed in terms of the mean velocity, V, and a correction coefficient, α.
The corrected velocity head becomes
α
V
2
Eqn, 4.26: α = Σ(v3dA)/V3A
2g
Some values of α (table 4.2 text book):
• α=2
(laminar pipe flow)
• α ≈ 1.06
(turbulent pipe flow)
• α ≈ 1.05
(turbulent flow in wide channel)
VVR 120 Fluid Mechanics
WHY THE ENERGY CORRECTION
COEFFICIENT α (MOMENTUM COEFFICIENT
β) OFTEN MAY BE OMITTED
1) Most engineering pipe flow problems involve turbulent flow in which
α is only slightly more than unity.
2) In laminar flow where α is large, velocity heads are usually
negligible when compared to the other Bernoulli terms
3) The velocity heads in most pipe flows are usually so small
compared to the other terms that inclusion of α has little effect
4) The effect of α tends to cancel since it appears on both sides of the
equation
5) Engineering answers are not usually required to an accuracy which
would justify the inclusion of α in the equation.
VVR 120 Fluid Mechanics
C14* Water is flowing.
The flow picture is
axisymmetric. Calculate
the flowrate and
manometer reading.
1
2
4
5
3
VVR 120 Fluid Mechanics
C23* Channel and gate are 1 m wide (normal
to the plane of the paper). Calculate q1, q2,
and Q3.
1
2
3