Fluid Dynamics

1
TOPIC 7

• The Momentum Balance

2
Course Organization

• Textbook deNevers Fluid Mechanics for Chemical
  Engineers
• Introduction (Chapter 1) / Dimensions, Units
• Fluid statics Fluid is at rest
• Fluid mechanics
• Fluid dynamics Fluid is moving
• Fluid statics (Chapter 2) Pressure, measurement
  of pressure, hydrostatic forces, buoyancy
• Fluid dynamics (Chapters 3-5, 7) Mass, energy
  and momentum balances
• Applications in Engineering (Chapters 6, 9, 11,
  12) Flow in pipes, turbomachines, flow over
  immersed bodies, flow through porous media
• Dimensional analysis and modeling (Chapter 13)

3
Recall The General Balance Equation
Creation Destruction Flow in Flow out
Accumulation
Rate Equation
Rate of Rate of Rate of
Rate of Rate of Creation Destruction
Flow in Flow out Accumulation
Applicable to any extensive property mass,
energy, entropy, momentum, electric charge
4
Conservation of Momentum

• For a mass, m, moving with a velocity V, the
  momentum is defined by
• Momentum m V ML/T

• Momentum Balance

Rate of accumulation of momentum into CV
Sum of forces acting on CV
Rate of momentum out of CV
Rate of momentum into CV
-


(7.1)
-


5
Conservation of Momentum
Equation (7.1) is a vector equation and can be
represented by three equations for each
coordinate direction
x-direction
y-direction
(7.2)
z-direction
6
Conservation of Momentum

• For one-dimensional and steady flow in the
  x-direction

(7.3)
7
Conservation of Momentum

• Convention The forces involved in the momentum
  balance are positive if they act on the CV and
  negative if exerted by the CV to the
  surroundings
• The forces acting on the fluid can be body forces
  and surface forces
• Body forces act on every mass element of the body
  and are proportional to the mass of the body.
  Examples are gravitational and electromagnetic
  forces
• Surface forces require a surface contact with the
  surroundings. Pressure is a surface force.

8
Example 1 Force on a Nozzle

• Determine the force exerted by a fluid on the
  walls of the horizontal nozzle shown in the
  figure.
• Use the following information to determine the
  numerical value of the force, if water (r1000
  kg/m3) is flowing through the nozzle. The
  following are given P11000 kPa (gauge
  pressure), P2Patm101.33 kPa, D125 cm, D25 cm,
  V250 m/s.

y
x
Now assume that the nozzle is attached to a fire
hose by a bolted flange. What is the force
tending to tear apart that flange?
9
Example 2 Force on a Pipe Bend

• Determine the force exerted by the fluid on the
  pipe bend shown in the figure.