Download Chapter8.Presentation.ICAM.Hydrostatics.Rev_April2015

NAZARIN B. NORDIN
[email protected]
What you will learn:
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Pascal’s law
Incompressibility of fluids
Pressure, force ratio
Archimedes principle
Density and relative density
Introduction to fluids
A fluid is a substance that can flow and
c onfor m t o the boundaries of a n y
container in which we put them.
e.g. water, air, glass.
• A fluid is any substance that can flow such as
a liquid or a gas.
• Fluids don’t have well defined shapes.
• A fluid takes on any shape to fit a container.
• The study of fluids can be divided into two
categories : hydrostatics and hydrodynamics
or fluid dynamics.
Basic properties of fluids
Density (mass per
unit volume) -   m / V
Pressure (force per
unit area) - P  F / A
Basic properties of fluids
Pressure (force per
unit area) - P  F / A
Notice that from
definition, pressure may
depend on direction.
However, this is not the
case for static fluids.
(why?).
Basic properties of fluids
Pressure (force per
unit area) - P  F / A
Unit of pressure:
1 pascal (Pa) = 1
Newton per square
meter.
1 atm. = 1.01 x 105 Pa
Fluids at rest
Pressure increases when we go “deeper”
into water – why?
F2  F1  mg,
F1  p1 A,
F2  p2 A,
m  A( y1  y2 )
Fluids at rest
Pressure of a fluid in static equilibrium
depends on depth only
p2  p1  g ( y1  y2 ),
or
p  p0  gh
Example
Which one of the four container + fluid has
highest pressure at depth h?
How about if (d) is move up (down)
by distance h?
What we will cover:
• Pascal’s Law:
For a confined fluid in a container, the change in pressure
will be transmitted without loss to every point of the liquid
and to the walls of the container
• Archimedes’ Principle:
Any body that is completely or partially submerged
in a fluid will experience an upthrust that is equal
to the weight of the fluid displaced by the body
1. Pascal’s Principle
• A change in the pressure applied to an
enclosed incompressible fluid is transmitted
undiminished to every portion of the fluid and
to the walls of the container as a direct
consequence of Newton’s Law.
3. Pressure and Pascal’s Law
Pressure applied to a
confined fluid increases
the pressure throughout
by the same amount.
In picture, pistons are at
same height:
F1 F2
F2 A2
P1  P2  


A1 A2
F1 A1
Ratio A2/A1 is called ideal
mechanical advantage
Example: Hydraulic level
• Applied force Fi 
change in pressure
p=Fi/Ai=Fo/Ao.
• Therefore output force
is Fo=FiAo/Ai.
• Therefore
• Fo > Fi if Ao > Ai
• How about work done?
The Manometer
P1  P2
P2  Patm   gh
An elevation change of
z in a fluid at rest
corresponds to P/g.
A device based on this is
called a manometer.
A manometer consists of
a U-tube containing one
or more fluids such as
mercury, water, alcohol,
or oil.
Heavy fluids such as
mercury are used if large
pressure differences are
anticipated.
Mutlifluid Manometer
For multi-fluid systems
Pressure change across a fluid
column of height h is P = gh.
Pressure increases downward, and
decreases upward.
Two points at the same elevation in a
continuous fluid are at the same
pressure.
Pressure can be determined by
adding and subtracting gh terms.
P2  1gh1  2 gh2  3 gh3  P1
Measuring Pressure Drops
C
Manometers are well-suited to measure
pressure drops across
valves, pipes, heat
exchangers, etc.
Relation for pressure
drop P1-P2 is obtained by
starting at point 1 and
adding or subtracting gh
terms until we reach point
2.
If fluid in pipe is a gas,
2>>1 and P1-P2= gh
The Barometer
PC   gh  Patm
Patm   gh
Atmospheric pressure is
measured by a device called a
barometer; thus, atmospheric
pressure is often referred to as
the barometric pressure.
PC can be taken to be zero
since there is only Hg vapor
above point C, and it is very
low relative to Patm.
Change in atmospheric
pressure due to elevation has
many effects: Cooking, nose
bleeds, engine performance,
aircraft performance.
4. Archimedes’ Principle
• When a body is fully or partially submerged
in a fluid, a buoyant force Fb from the
surrounding fluid acts on the body. The force
is directed upward and has a magnitude
equal to the weight mfg of the fluid that has
been displaced by the body.
Archimedes’ Principle
• Buoyant force – upward
force in liquid because of
increasing pressure in
liquid as one goes down
below the surface.
• (a) a hole in water.
Notice that the hole is in
static equilibrium if it is
filled with water.
Archimedes’ Principle
• (a) a hole in water.
Notice that the hole is in
static equilibrium if it is
filled with water.
• Therefore the upward
force = mfg, mf = mass of
displaced water.
Archimedes’ Principle
• (b) The hole in water is
replaced by a solid with
the same shape.
• Since nothing changes in
water, therefore the
upward force = mfg, mf =
mass of displaced water
= buoyant force
Archimedes’ Principle
• (c) The solid in water is
replaced by a piece of
wood with mw < mf..
• In this case the wood
float on the surface with
Fb=mwg.
Archimedes’ Principle
• Question: Imagine a large sphere of water
floating in outer space. The sphere of water
is formed under its own gravity. Is there any
buoyant force if an object enters this sphere
of fluid?
Examples of Archimedes
Principle
The Golden Crown of Hiero II, King of Syracuse
• Archimedes, 287-212 B.C.
• Hiero, 306-215 B.C.
• Hiero learned of a rumor where the
goldsmith replaced some of the gold
in his crown with silver. Hiero asked
Archimedes to determine whether
the crown was pure gold.
• Archimedes had to develop a
nondestructive testing method
The Golden Crown of Hiero II, King of Syracuse
• The weight of the crown and nugget
are the same in air: Wc = cVc = Wn =
nVn.
• If the crown is pure gold, c=n
which means that the volumes must
be the same, Vc=Vn.
• In water, the buoyancy force is
B=H2OV.
• If the scale becomes unbalanced,
this implies that the Vc ≠ Vn, which in
turn means that the c ≠ n
• Goldsmith was shown to be a fraud!
3. PRESSURE
• Pressure is the quantity that is related to the force acting on
the walls of the balloon and is defined as the normal force per
unit area.
• If F is the force perpendicular to the surface area A, the
pressure P is therefore
F
P
A
• The pressure at a point in a fluid depends on the depth.
Greater depths result in greater pressures
Fluids in Motion
• An ideal fluid is one that
(i) flows smoothly,
(ii) is non-viscuous,
(iii) is incompressible,
(iv) is irrotational.
• The path of steady flow can be visualized using streamlines.
• Under steady-state flow conditions, for a given time interval,
the volume of liquid flowing into the tube must equal the
volume of liquid flowing out of a tube.This is known as the
Continuity Equation.
4. Bernoulli’s Equation
i.
Bernoulli’s Equation relates the elevation y, speed v and
pressure P of a fluid at any point in a tube.
ii. According to Bernoulli’s Equation:
for an inviscid flow of a non-conducting fluid, an increase in
the speed of the fluid occurs simultaneously with a decrease
in pressure or a decrease in the fluid's potential energy. The
principle is named after Daniel Bernoulli who published it in
his book Hydrodynamica in 1738.
iii. However, Bernoulli’s Equation is not applicable to viscous
fluids
Bernoulli’s Equation
• Bernoulli’s Equation is
a consequence of
conservation of
energy in steady flow.
W  K ;
1
1
2
K  mv2  mv12
2
2
1
 V (v22  v12 )
2
Bernoulli’s Equation
• Bernoulli’s Equation is
a consequence of
conservation of
energy in steady flow.
W  Wg  WP ;
Wg  ( V ) g ( y2  y1 )
W p   p2 V  p1V
Bernoulli’s Equation
• Adding together, we
obtain
1 2
1 2
p1  v1  gy1  p2  v2  gy2
2
2
or
1 2
p  v  gy  c
2
(Bernoulli’s
Equation)
Water pressure in a home (Bernoulli’s Principle II)
• Consider this example:
Example
• What is the speed v of
the water emerging
from the hole?
• Show that v2=2gh
(same as free fall)
The continuity equation for
flowing liquids
• The continuity equation – conservation of
mass in a incompressible liquid flow.
V  A1v1t  A2 v2 t
or
A1v1  A2 v2
v = velocity of fluid
flowing through
area A in the tube
Example
• What is the volume flow rate of water if
Ao=1.2cm2, A=0.35cm2 and h=45mm.
A0 v0  Av
v  v  2 gh
2
2
0
2 ghA2
 v0 
 28.6cm / s.
2
2
A0  A
RV  A0 v0  34cm 3 / s.
5. DENSITY AND RELATIVE DENSITY
•The density is an important factor that
determines the behaviour of a fluid.
•The density of a fluid  is defined as the
mass m per unit volume V:
m

V
•The SI unit for the density is kg / m
3
5. DENSITY AND RELATIVE DENSITY
•Relative density, or specific gravity is the ratio of the density
(mass of a unit volume) of a substance to the density of a
given reference material.
•The relative density of a fluid
is defined as the density
of a substance over the density of water at 40C at standard
temperature and pressure:
𝜌𝑟 =
𝜌𝑠𝑢𝑏𝑠𝑡𝑎𝑛𝑐𝑒
𝜌𝑤𝑎𝑡𝑒𝑟 𝑎𝑡 4 deg 𝐶𝑒𝑙𝑐𝑖𝑢𝑠 𝑎𝑛𝑑 𝑆𝑇𝑃
•There is no SI unit for relative density.
Densities of common substances—Table 8.1
Table 8.1
QuickCheck 8.1
A piece of glass is broken
into two pieces of different
size. How do their densities
compare?
A. 1 > 3 > 2.
B. 1 = 3 = 2.
C. 1 < 3 < 2.
Slide 15-23
QuickCheck 8.1
A piece of glass is broken
into two pieces of different
size. How do their densities
compare?
A. 1 > 3 > 2.
B. 1 = 3 = 2.
C. 1 < 3 < 2.
Density characterizes the substance itself,
not particular pieces of the substance.
Slide 15-24
Liquids in Hydrostatic Equilibrium
 No!
 A connected liquid in hydrostatic equilibrium rises to
the same height in all open regions of the container.
Slide 15-40
QuickCheck 8.2
What can you say about
the pressures at points
1 and 2?
A. p1 > p2.
B. p1 = p2.
C. p1 < p2.
Slide 15-41
QuickCheck 8.2
What can you say about
the pressures at points
1 and 2?
A. p1 > p2.
B. p1 = p2.
C. p1 < p2.
Hydrostatic pressure is the same at all points on a
horizontal line through a connected fluid.
Slide 15-42
Liquids in Hydrostatic Equilibrium
 No!
 The pressure is the same at all points on a horizontal line
through a connected liquid in hydrostatic equilibrium.
Slide 15-43
QuickCheck 8.3
An iceberg floats in a
shallow sea. What can
you say about the
pressures at points
1 and 2?
A. p1 > p2.
B. p1 = p2.
C. p1 < p2.
Slide 15-44
QuickCheck 8.3
An iceberg floats in a
shallow sea. What can
you say about the
pressures at points
1 and 2?
A. p1 > p2.
B. p1 = p2.
C. p1 < p2.
Hydrostatic pressure is the same at all points on a
horizontal line through a connected fluid.
Slide 15-45