Fluid Statics

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TOPIC 2

• Fluid Statics

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Fluid Statics

• The word statics is derived from Greek word
  statikos motionless
• For a fluid at rest or moving in such a manner
  that there is no relative motion between
  particles there are no shearing forces present
  Rigid body approximation

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Definition of Pressure
Pressure is defined as the amount of force
exerted on a unit area of a substance P F / A
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Pascals Laws

• Pascals laws
• Pressure acts uniformly in all directions on a
  small volume (point) of a fluid
• In a fluid confined by solid boundaries, pressure
  acts perpendicular to the boundary it is a
  normal force.

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Direction of fluid pressure on boundaries
Furnace duct
Pipe or tube
Heat exchanger
Pressure is due to a Normal Force (acting
perpendicular to the surface) It is also called a
Surface Force
Dam
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Absolute and Gauge Pressure

• Absolute pressure The pressure of a fluid is
  expressed relative to that of vacuum (0)
• Gauge pressure Pressure expressed as the
  difference between the pressure of the fluid and
  that of the surrounding atmosphere.
• Usual pressure gauges record gauge pressure. To
  calculate absolute pressure
• Pabs Patm Pgauge

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Units for Pressure
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Pressure distribution for a fluid at rest

• We will determine the pressure distribution in a
  fluid at rest in which the only body force acting
  is due to gravity
• The sum of the forces acting on the fluid must
  equal zero
• Consider an infinitesimal rectangular fluid
  element of dimensions Dx, Dy, Dz

z
y
x
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Pressure distribution for a fluid at rest

• Let Pz and PzDz denote the pressures at the base
  and top of the cube, where the elevations are z
  and zDz respectively.
• Force at base of cube Pz APz (Dx Dy)
• Force at top of cube PzDz A PzDz (Dx Dy)
• Force due to gravity m gr V g r (Dx Dy Dz) g
• A force balance in the z direction gives

For an infinitesimal element (Dz?0)
?
(2.1)
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Incompressible fluid

• Liquids are incompressible i.e. their density is
  assumed to be constant
• When we have a liquid with a free surface the
  pressure P at any depth below the free surface
  is

(2.2)
where Po is the pressure at the free surface
(PoPatm) and h zfree surface - z
(2.3)

• By using gauge pressures we can simply write

(2.4)
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Example Pressure in an Oil Storage Tank

• The figure below shows a schematic of a crude
  oil storage tank. What is the absolute pressure
  at the bottom of the cylindrical tank, if it is
  filled to a depth of H with crude oil, with its
  free surface exposed to the atmosphere? The
  specific gravity of the crude oil is 0.846. Give
  the answers for
• H5.0 m (pressure in Pa and bar).
• H15.0 ft (pressure in lbf / in2)
• What is the purpose of the surrounding dike?

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Compressible fluid

• Gases are compressible i.e. their density varies
  with temperature and pressure r P M /RT
• For small elevation changes (as in engineering
  applications, tanks, pipes etc) we can neglect
  the effect of elevation on pressure
• In the general case start from Eq. (2.1)

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Example

• Compute the atmospheric pressure at an altitude
  of 5000 m if the pressure at sea level is 101.3
  kPa by the following methods a) assume air of
  constant density r1.24 kg/m3 and b) assuming
  that the density of air changes with altitude,
  but temperature remains constant.

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Hydrostatic forces on plane surfaces

• Case 1 Horizontal surface exposed to a gas

• Pconstant everywhere
• F P . A

• Case 2 Horizontal surface exposed to a liquid

• Pconstant along the horizontal surface
• F P . A

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Example

• The crude oil storage tank shown in page 2.11 has
  a flat, horizontal circular roof 150 ft in
  diameter. The atmospheric pressure is 14.7 psia.
  What force does the atmosphere exert on the roof?
• A layer of rainwater 4 in deep collects on the
  roof of the tank. What net pressure force does it
  exert on the roof of the tank? (typical values of
  density of water can be found in the back cover
  of your textbook)

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Hydrostatic forces on plane surfaces

• Case 3 Vertical surface exposed to air

• Pressure varies linearly with height (see also
  equation 2.4) Prgh
• However, because r of gases is very low, the
  dependence is very week
• Therefore we can assume that Pconstant
  everywhere
• P F . A

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Hydrostatic forces on plane surfaces

• Case 4 Vertical surface exposed to liquid

• Example The lock gate of a canal is
  rectangular, 20 m wide and 10 m high. One side is
  exposed to the atmosphere and the other side to
  the water. What is the net force on the lock gate?

• Here the pressure varies linearly with depth (see
  also equation 2.4) Prgh

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Vertical plane surfaces

• For an infinitesimal area dA the normal force due
  to the pressure is
• dF p dA
• Find resultant force acting on a finite surface
  by integration

• For a vertical rectangular wall F ½ r g W
  H2

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Vertical surfaces - General
For surface of arbitrary shape we can write
By definition the centroid of the surface, hc is
Therefore F r g hC A (2.5)

• The force exerted on a submerged plane surface is
  given by the product of the area and the pressure
  at the centroid. The location of the centroid is
  known for several geometries

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Centroid Location for Common Shapes
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Example 1 Centroid Method

• Redirive the expression for the force exerted on
  the lock gate shown in page 2.17, by using the
  centroid method

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Example 2 Centroid Method

• A vertical wall, shaped as an inverse triangle,
  with H3m high and W2m wide at the top is
  submerged in water. The wall is just level with
  the surface of the water upstream. Determine the
  force exerted by the water on the wall, by using
  the centroid method.

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Buoyancy

• Laws of buoyancy discovered by Archimedes
• A body immersed in a fluid experiences a vertical
  buoyant force equal to the weight of the fluid it
  displaces
• A floating body displaces its own weight in the
  fluid in which it floats

Free liquid surface
F1
h1
The upper surface of the body is subjected to a
smaller force than the lower surface ? A net
force is acting upwards
H
h2
F2
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Buoyancy
The net force due to pressure in the vertical
direction is FB F2- F1 (Pbottom - Ptop)
(DxDy) The pressure difference is Pbottom
Ptop r g (h2-h1) r g H From (2.6) FB r g H
(DxDy) Thus the buoyant force is FB r g V
(2.6)
where r the fluid density
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Example

• Consider a solid cube of dimensions 1ft x 1ft x
  1ft (0.305m x 0.305m x 0.305m). Its top surface
  is 10 ft (3.05 m) below the surface of the
  water. The density of water is rf1000 kg/m3.
• Consider two cases
• The cube is made of cork (rB160.2 kg/m3)
• b) The cube is made of steel (rB7849 kg/m3)
• In what direction does the body tend to move?

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Measurement of Pressure

• The atmospheric pressure can be measured with a
  barometer.
• For mercury barometers atmospheric pressure
  (101.33kPa) corresponds to h760 mmHg ( 29.2 in)
• If water is used h 10.33 m H2O ( 34 ft)

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Measurement of Pressure

• Manometers are devices in which one or more
  columns of a liquid are used to determine the
  pressure difference between two points.
• U-tube manometer
• Inclined-tube manometer

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Pascals principle (The hydrostatic paradox)

• From equation (2.3) the pressure at a point in a
  fluid depends only on density, gravity and depth.
  
• The pressure in a homogeneous, incompressible
  fluid at rest depends on the depth of the fluid
  relative to some reference plane, and it is not
  influenced by the size or shape of the tank or
  container

Fluid is the same in all containers
h
Pressure is the same at the bottom of all
containers
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Example 1 Manometer
A
Manometric fluid, density rM
Fluid, density rT
Find the gage pressure at point A
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Example 2 Measurement of Pressure Differences
PA
PB
Find the pressure difference PA-PB
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Measurement of Pressure

• Mechanical and electronic pressure measuring
  devices
• When a pressure acts on an elastic structure it
  will deform. This deformation can be related to
  the magnitude of the pressure.
• Bourdon pressure gage
• Pressure transducers convert pressure into an
  electrical output
• Strain-gage pressure transducers are suitable for
  rapid changes in pressure and cover big ranges of
  pressure values

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Summary

• The behavior of static fluids has been examined
• The pressure distribution in a fluid at rest has
  been determined
• Specific applications have been considered,
  including manometry, forces on plane submerged
  surfaces and buoyancy.
• Our next task is to examine the behavior of
  fluids in motion.