FLUID STATICS AND
MANOMETRY
Hydrostatics
• Fluid statics (or Hydrostatics) is the branch of fluid mechanics
which studies systems in which the fluid is at rest (i.e., fluid
velocity is zero or is uniform and constant relative to an inertial
reference frame.
Note: Fluids at rest experience no shear stress.
Pascal’s principle: The pressure in a fluid at rest is the same at all
points with the same elevation.
Basic Principle
The basic property of a static fluid is pressure. Pressure is familiar as
a surface force exerted by a fluid against the walls of its container.
It also exists at every point within a volume of fluid.
• Consider a differential volume of fluid at rest whose volume is dx
dy dz
• Consider the x-direction:
P
x dx
P + P
For fluid at rest:
 Fx  0
P 

dx  dydz  0
 Fx  Pdydz  P 
x


P
0
x
• The same goes for y-direction. Therefore, pressure does not vary in x- or y-direction.
• Consider the z-direction:
For fluid at rest:
 Fz  0
P
dz  dxdy  gdxdydz  0
 F  P dxdy   P 
z


z


This suggests that pressure varies in vertical direction. This is the equation
that represents the hydrostatic principle (Pascal’s principle).
The Hydrostatic Principle
Assumptions:
• There is no shear stress along z, i.e. static fluid.
• Gravity is the only force acting on the fluid.
• z-axis direction is defined vertical and upward
• Since P does not vary along x- and y-direction:
• Here, gc is the Newton’s law correction factor equal to
1 kg.m N-1 s-2 or 32.174 ft.lbm lbf-1 s-2.
• For incompressible fluids (constant ), integrating the equation
P  
g
z
gc
• For compressible fluids at constant temperature, assuming ideal
gas law applies, integrating the equation
P
g M
ln 2  
P1
gc RT z  z 
2
1
• If the temperature is not constant, the following equation in
integral form applies:
P
g M z dz
ln 2  

P1
gc R 0 T(z)
• In some part of our atmosphere, T varies with height. In
troposphere (up to  11 km):
T  To  z
• where  = - 0.00651 K m-1 (for air).
• The variation of pressure with height is given by the following
equation:
P
g M
T  z 2
ln 2  
ln o
P1
gc R To  z1
• In stratosphere (about 11 to 32 km), T is constant at  -550C.
• Pressure Measurement
Bourdon gauge
normally used to measure gauge pressure in
pipeline (on-line measurement)
Barometer
used to measure atmospheric pressure
Manometer
used to measure unknown pressure, or
differences of two unknown pressures
• Manometry
• Manometry is a common and accurate method of measuring
relatively low fluid pressure by causing the pressure to raise a
column of liquid.
U-tube
Is
an
extension
of
the
piezometer v.tiere a double limb tube
· .....,,
T'
or "\.J" tube is used. A U-tube may be
vertical, inverted, or indined.
tube
manometer
The U
measures
difference
between
pressure
and
the
the
the
absolute
••
•
atmospheric
pressure. The difference is called the
gage pressure and is frequently used in
pressure measurement. The pressure
w111•• .,,. . .. . , .,... 11, p
11111
A u-tube manometer
of both liquids and gases can be
measured using the same U-tube.
A U-tube is connected as in Figure 5.2a and filled with a fluid called the
manometric Duid.
c
D
•
h
•
-=.- G o s-_or- =:- - -•
--- --li u id
- - -- - ------== -
--=- A=
••
Va c u u m
••
- - - (b
)
(e )
.-····-··,,.
0 1!h e r
D i f f e r e n t i e l·
Perry's
-------- .. .·-·-....
(c)
( F ig .
C H E
11 0 - . . 0
U - 1. u b e
u - ·t u b e
m an om eter
H a nd b o o k );
P e r r y 's
c o n f i g u r a t io n s :
C H E
(b)
U - tu b e
H e n db o o k) ;
p r e s s u r e d i f f e r e n c e m e a s u re m e n t b y
m a n o m e te r
110 - 9
(Fig_
c lo s e d
a
(a )
(c )
U -t u b e
• Consider simple U-tube manometer shown
• Applying Pascal’s principle with A as the reference point:
P P  
A
B
g
h 0
gc
1
• Applying Pascal’s principle with D as the reference point:
PD  PC  man
g
h2  0
gc
density of gas is negligible compared with density of fluid in the manometer
• Subtracting equation 2nd equation from the first and knowing that
PC = PB and PD = Patm:
PA PD 
g
g
h1 man h2 0
gc
gc
g
PA Patm  manh2 h1
gc
Rules of thumb for multiple-tube manometers:
• Any two points at the same elevation in a continuous volume of the same
liquid are at the same pressure.
• Pressure increases as one goes down a liquid column.