Download CVE 240 – Fluid Mechanics

CHAPTER 6
MOMENTUM PRINCIPLE
Dr . Ercan Kahya
Engineering Fluid Mechanics 8/E by Crowe, Elger, and Roberson
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
MOMENTUM PRINCIPLE
F  ma
Newton’s second law; for a single particle
F 
d ( mV )
dt
F 
d(Mom sys )
dt
in terms of momentum; for a single particle
For a system of particles – Lagrangian
form
Reynolds Transport Theorem for momentum
d (Momsys )
dt
d
  vdV   vV .dA
dt cv
cs
b: intensive property is momentum
per unit mass = m v / m = v
Combining the last two equations:
d
F   vdV   vV .dA
dt cv
cs
Momentum Principle
MOMENTUM PRINCIPLE
d
F   vdV   vV .dA
dt cv
cs
• This equation states that the sum of the external forces acting
on the material in the control volume equals the rate of
momentum change inside the control volume (aka momentum
accumulation) plus net rate at which momentum flows out of
the control volume.
• If there is no momentum accumulation;
F   vV .dA
cs
Zero momentum accumulation is
common for many problems in fluid
mechanics
Reynolds Transport Theorem: Simplified form
If the flow crossing the control surface occurs through a number of inlet and
outlet ports, and the velocity v is uniformly distributed (constant) across each
port; then


d
F   vdV  cs vo mo  cs vi mi
dt cv
In Cartesian coordinate (x,y,z) system, the component form of the momentum
equation in x-direction:


d
Fx   vx dV  cs vox mo  cs vix mi
dt cv
Force Terms
FORCE DIAGRAMS
Forces associated with flow in a pipe:
(a) pipe schematic,
(b) control volume situated inside the pipe,
(c) control volume surrounding the pipe.
In (b): The fluid within the control volume has been isolated from its
surroundings and the effects of the surroundings are shown as forces.
Forces in z direction: pressure, shear, and weight
In (c): The control volume cuts through the pipe wall.
Forces in z direction: pressure, tension (F1, F2), and weight (fluid +wall)
FORCE TYPE
• Body Force: A force that acts on mass elements within the body
and acts at a distance without any physical contact.
– Gravitational
– Electrostatic
– Magnetic
• Surface Force: A force that requires physical contact, acting at
the control surfaces
– All forces except weight in figure (b) and (c)
– For example: hydrostatic pressure – pressure integrated over the
area of control surface
Momentum Equation: Useful Form for Steady Flow


F   m o v ox   m i v ix
This is the key equation to Chapter 6.
Sign convention is important !
For each flow stream: is it going
“in” or “out” of the CS?
Note the flow direction, and
include that in the velocity term
y
In
x
Out
Systematic Approach
• Problem setup
• Select a control volume. Select coordinate axes. Select an inertial reference
frame.
• Identify governing equations (scalar vs. vector), and the other equations
(Bernoulli, continuity) may also be applied.
• Force analysis and diagram
• Sketch body forces on the force diagram
• Sketch surface forces on the force diagram
• Momentum analysis and diagram
• Evaluate the momentum accumulation term. If the flow is steady and other
materials in the CV are stationary, it is zero.
• Sketch momentum flow vectors on the momentum diagram. For uniform
velocity, each vector is d mV / dt
Typical Momentum Applications
Fluid Jets
Nozzles
Vanes
Pipe Bends
EXAMPLE : Momentum Application
(Q 6.8)
A 15 m/s jet of water (diameter 30 mm) is filling a tank. The tank has a mass of 5
kg, and contains 20 liters of water as shown. The water temp is 15 deg C.
Find:
‐ Force acting on the stop block.
‐ Force acting on the bottom of the tank
SOLUTION
SOLUTION………….cont.
Moving Control Volumes
d
F   vdV   vV .dA
dt cv
cs
Important reminder:
V : fluid velocity relative to the control surface where the
flow crosses the surface.
It is always measured w.r.t. the control surface
because it relates to the mass flux across the surface.
v : velocity used to define momentum “mv” of any fluid
particle of mass “m” in the system & is relative to an
inertial reference frame. This frame does not rotate
and can be either stationary or moving at a constant
velocity.
It is evaluated at the control surface w.r.t. the inertial
reference frame selected.
Moving Control Volumes
• Up to this point, we have been going through the applications of the
momentum equation involving a stationary control volume (CV)
• In some problems, to attach CV to a moving body is more useful...
• For example, jet impinging on moving block (see Example 6.10)
• Inertial reference frame on the block
• Vx = 0
zero accumulation term
• Vix = Vj – Vb at station 1
• Vox = 0
at station 2
• Flow is steady w.r.t. the block, so
• Mass flow rate = ρ A (Vj – Vb)
EXAMPLE: Momentum Application-Vane (Q6.29)
• A horizontal jet strikes a vane that is moving at a speed
vv = 7 m/s. Diameter of the jet is 6 cm. Speed of the fluid jet
is 20 m/s, relative to a fixed frame. What components of
force are exerted on the vane by the water in the x and y
directions? Assume negligible friction between the water
and the vane.
SOLUTIONS
SOLUTIONS cont……..
Class Exercises: (Problem 6.26)
Class Exercises: (Problem 6.55)