Download Bernoulli Equation

The Bernoulli Equation
• It is an approximate relation between pressure,
velocity and elevation
• It is valid in regions of steady, incompressible flow
where net frictional forces are negligible
• Viscous effects are negligible compared to inertial,
gravitational and pressure effects.
• Applicable to inviscid regions of flow (flow regions
outside of boundary layers)
• Steady flow (no change with time at a specified
location)
Steady flow
• The value of a quantity may change from one location to
another. In the case of a garden hose nozzle, the velocity
of water remains constant at a specified location but it
changes from the inlet to the exit (water accelerates along
the nozzle).
Acceleration of a Fluid Particle
• Motion of a particle in terms of
distance “s” along a streamline
• Velocity of the particle, V = ds/dt,
which may vary along the streamline
• In 2-D flow, the acceleration is
decomposed into two components,
streamwise acceleration as, and
normal acceleration, an.
V2
an 
R
• For particles that move along a straight path, an =0
Fluid Particle Acceleration
• Velocity of a particle, V
(s, t) = function of s, t
V
V
dV 
ds 
dt
s
t
• Total differential
dV V ds V
or


dt
s dt t
• In steady flow,
V
 0;andV  V ( s )
t
• Acceleration,
dV V ds V
dV
as 


V V
dt
s dt s
ds
Derivation of the Bernoulli Equation (1)
• Applying Newton’s second law of conservation of linear
momentum relation in the flow field
dV
PdA  ( P  dP)dA  W sin   mV
ds
m  V   dAdsisthemass
W=mg= gdAdsistheweight of the fluid
sin =dz/ds
Substituting,
dz
dV
-dpdA -  gdAds   dAdsV
ds
ds
dp   gdz  VdV ,
Canceling dA from each term and simplifying,
1
Note V dV= d (V 2 ), and divding by 
2
dp 1
 d (V 2 )  gdz  0
 2
Derivation of the Bernoulli Equation (2)
• Integrating
For steady flow
dp V 2
   2  gz  constant (along a streamline)
For steady incompressible flow,
p V2
  gz  constant (along a streamline)
 2
Bernoulli Equation
• Bernoulli Equation states
that the sum of kinetic,
potential and flow (pressure)
energies of a fluid particle is
constant along a streamline
during steady flow.
• Between two points:
p1 V12
p2 V2 2

 gz1 

 gz2 or,
 2

2
p1 V12
p2 V2 2

 z1  
 z2
 2g
 2g
p
V2
 pressure head;
 velocity head, z=elevation head

2g
Example 1
Figure E3.4 (p. 105) Flow of
water from a syringe
Example 2
• Water is flowing from a hose attached to a water main at
400 kPa (g). If the hose is held upward, what is the
maximum height that the jet could achieve?
Example 3
• Water discharge from a large tank. Determine the water
velocity at the outlet.
Limitations on the use of Bernoulli
Equation
• change in flow conditions
• Frictional effects can not be neglected in long and narrow
flow passage, diverging flow sections, flow separations
• No shaft work