Transcript Hydrostatic Equilibrium
CHAPTER 2
Fluid Statics and Its Applications
Nature of fluids
Hydrostatic Equilibrium
Applications of fluid statics
Nature of fluids
A fluid is a substance that does not permanently resist distortion.
During the change in shape, shear stresses exist, the magnitudes of which depend upon the viscosity of the fluid and the rate of sliding
Fluids include
liquid ,
gas
and solid particles suspended in liquid and gas or slurry
Fluids also can be divided as
Incompressible
——the density changes only slightly with moderate changes in temperature and pressure
Compressible
——the changes in density caused by temperature and pressure are significant (Pressure concept : the pressure at any point in the fluid is independent of direction)
Hydrostatic Equilibrium
There is a vertical column of fluid shown in Fig.2.1
Three vertical forces are acting on this volume: (1)the force from pressure p acting in an upward direction , which is pS ; (2) the force from pressure p+dp acting in a downward direction , which is
(
p+dp)S ; (3)the force of gravity acting downward, which is gρsdz
p
+
dp p g
Figure2.1 Hydrostatic equilibrium
Then
pS
(
p
0 (2.1) After simplification and division by S ,Eq.(2.1) becomes
dp
0 (2.2) Integration of Eq.(2.2) on the assumption that density is constant gives
p
gZ
const
(2.3)
Between the two definite heights Z a and Z b shown in Fig.2.1,
p b
p b
a
Z b
) (2.4) Equation (2.3) expresses mathematically the condition of hydrostatic equilibrium.
Gauge pressure, absolute pressure and vacuum
The relationship between gauge pressure and absolute pressure P(gauge)=P(absolute)-P(atmosphere) The relationship between vacuum and absolute pressure P(vacuum)=P(atmosphere)-P(absolute) Or P(vacuum)=- P(gauge)
The reading in the gauge is 1.5 kgf /cm2 = =(?)N/m2, and the reading of the vacuum gauge is 736 mmHg = ( )m H2O .If the atmospheric pressure is 1 atm, what happens to the above cases in absolute pressure?
or
Barometric equation
For an ideal gas , the density and pressure are related by the equation = pM RT (2.5) Substitution from Eq.(2.5)intoEq.(2.2)gives = pM RT
dp
p gM RT dZ
0 (2.6)
on the assumption that T is constant,gives ln
p b p a
gM RT
(
Z b
Z a
) or
p b p a
exp
gM RT
(
Z b
Z a
) (2.7) Equation(2.7)is known as the barometric equation.
Hydrostatic equilibrium in a centrifugal field
In a rotating centrifuge a layer of liquid is thrown outward from the axis of rotation and is held against the wall by centrifugal force.
The free surface of the liquid takes the shape of a paraboloid of revolution.
The rotational speed is so high and the centrifugal force is so much greater than the force of gravity that the liquid surface is virtually cylindrical and coaxial with the rotation.
The situation in shown in Fig.
dr r 2 r 1 r
The entire mass of liquid indicated in Figure is rotating as a rigid body, with no sliding of layer of liquid over another.
Under these conditions the pressure distribution in the liquid may be found from the principles of fluid static.
• The pressure drop over any ring of rotating liquid is calculated as follows.
The volume element of thickness
d r
at a radium
r
.
dF
2
rdm
If
ρ
is the density of the liquid and
b
the breadth of the ring.
dm
rbdr
Eliminating
dm
gives
dF
2 2 2
r dr
The change in pressure over the element is the force exerted by the element of liquid, divided by the area of the ring.
dp
dF
2
rb
2
rdr
The pressure drop over the entire ring is
p
2
p
1
r
1
r
2 2
rdr
Assuming the density is constant and integration gives
p
2
p
1 (
r
2 2
r
1 2 ) 2 (2.8)
Applications of fluid statics
Manometer (pressure gauge)
The manometer is an important device for measuring pressure differences.
U tube manometer (or reverse U tube)
Inclined manometer
Differential manometer
U tube manometer
It is the simplest form of manometer.
A pressure
p a
arm of
U
is exerted in one tube and a pressure
p b
in the other.
As a result of the difference in pressure, the meniscus in one branch of the tube is higher than p a p b that in the other .
ρ B z m
3 4 Vertical distance between the
R m
1 2 two meniscuses
R m
may be used to measure the difference
ρ A
in pressure.
The pressure at the point 1 is
p
1
p a
m
R m
)
B
The pressure at the point 2 is
p
2
p b
gz m
B
gR m
A
p
1 is equal to
p
2 for the continuous fluid at the same level, thus
p a
m
R m
)
B
p b
gz m
B
gR m
A
Simplification of this equation gives
p a
p b
gR
(
m
A
B
) (2.10) Note that this relationship is independent of the distance
z m
, and of the dimensions of the tube, provided that pressure
pa
and
pb
are measured in the same horizontal plane.
If fluid
B
is a gas,
ρ B
compared to
ρ A
is usually negligible and may be omitted from Eq. (2.10)
Inclined manometer
Used for measuring small differences in pressure.
By making
α
small, the magnitude of
R
m is multiplied into a long distance
R
1 , and large reading becomes equivalent to a small pressure difference
p a
p b
g
A
B
R
1 sin (2.11)
Differential manometer
P 2 P 1 B C A
Example: H2O flows through the pipe as shown in Fig. A U-tube manometer is used to measure the pressure P in the pipe. If the atmosphere pressure pa is 1 atm, R and h of mercury and water columns are 0.1 and 0.5 m, respectively, what is pressure P in the pipe, N/m2?
p h p a R
A A' 图 1.11