Vorticity in 3D –
Energy Equation
Vorticity in 2D –
Angular Velocity –
𝜔=
𝜁
2
Mass conservation
𝐦̇ = 𝛒𝐯𝐀 = 𝛒𝐕̇ =
𝐕̇
𝐯
Energy Analysis of Steady Flows
(P = Pa, 𝐖̇ = 𝐖𝐚𝐭𝐭𝐬) –
Steady Flow –
Acceleration Field
𝐚(𝐱, 𝐲, 𝐳, 𝐭) =
ⅆ𝐕 𝛛𝐯
𝛛𝐯
𝛛𝐯
=
+𝐮
+𝐯
ⅆ𝐭
𝛛𝐭
𝛛𝐱
𝛛𝐲
𝛛𝐯
+𝐰
𝛛𝐳
Incompressible Flow –
Streamlines
ⅆ𝐫
𝐕
ⅆ𝐲
ⅆ𝐱
=
=
ⅆ𝐱
𝐮
𝐯
𝐮
=
ⅆ𝐲
𝐯
=
ⅆ𝐳
𝐰
Bernoulli Equation
𝟏
𝟏
∫ 𝐯 ⅆy = ∫ 𝐮 ⅆx
OR
𝑷𝟏 𝒗𝟐𝟏
𝑷𝟐 𝒗𝟐𝟐
+
+ 𝒛𝟏 =
+
+ 𝒛𝟐
𝝆𝒈 𝟐𝒈
𝝆𝒈 𝟐𝒈
OR
Momentum Conservation
momentum analysis of flow systems? What
type of flow is it significant and must it be
considered in analysis? A: The momentumflux correction factor enables us to express
the momentum flux in terms of the mass
flow rate and mean flow velocity and this is
more significant for Laminar Flow
Q: Write the momentum equation for steady
one-dimensional flow for the case of no
external forces and explain the physical
significance of its terms. A: Since there is no
external force, it obeys law of conservation
of momentum.
Kinetic Energy Correction Factor
(𝜶) –
Steady Flow -
Q: In the application of the momentum
equation, explain why we can usually
disregard the atmospheric pressure and work
with gage pressures only. A: Since the
atmospheric pressure acts in all directions,
and its effect cancels out in every direction
so we can disregard the atmospheric pressure
and work with gage pressures only.
Mass Flow rate across an inlet or outlet -
Momentum Flow rate across a uniform inlet
or outlet –
OR
Q: Two firefighters are using identical water
hoses and nozzles, one is holding the hose
straight while the other holds it backward,
Which fighter will experience a greater
force? A: The firefighter who holds the hose
backward will experience a greater reaction
force because of the momentum effect.
Q: A rocket in space (no friction or
resistance to motion) can expel gases relative
to itself at some high velocity V. Is V the
upper limit to the rocket’s ultimate velocity?
A: No, in the presence of external force
(Fuel), the rocket will continue to accelerate,
regardless of how that force is generated.
Thus, its velocity will also continue to
increase.
Q: In a given location, would a helicopter require
more energy in summer or winter to achieve a
specified performance? Explain. A: The helicopter
needs more energy to hover in summer than winter
since summer has lower density.
One inlet and One outlet -
Along x-coordinate -
Chapter 6 Concept Questions
Q: How do surface forces arise in the
momentum analysis of a control volume?
How can we minimize the number of surface
forces exposed during analysis? A: Surface
forces arise as the control volume is isolated
from its surroundings for analysis. To
minimize surface forces in an analysis, we
must choose the control volume wisely.
Q: What is the importance of the
momentum-flux correction factor in the
Q: In a given location, would a helicopter require
more energy in summer or winter to achieve a
specified performance? Explain. A: With increase
of plate velocity, the relative velocity decreases,
hence the impulse force decrease, which further
reduces the acceleration of the plate
Q: A horizontal water jet of constant velocity V
from a stationary nozzle impinges normally on a
vertical flat plate that rides on a nearly frictionless
track. As the water jet hits the plate, it begins to
move due to the water force. What is the highest
velocity the plate can attain? A: The maximum
velocity is that of the jet velocity
Q: A horizontal water jet from a nozzle of
constant exit cross section impinges normally on
a stationary vertical flat plate. A force F is
required to hold the plate against the water
stream. If the water velocity is doubled, will
the necessary holding force also be doubled?
A: When velocity is doubled the force is
quadrupled.