pressure and fluid statics

Er Ranjit singhAsstt. Prof.D.M.E.S.G.G.S.W.U., Fatehgarh Sahib

PRESSURE AND FLUID STATICS

Chapter 2

Pressure is defined as the amount of force exerted on a unit area of a substance:

PRESSURE

Pam

N

area

forceP

2

MEASURING PRESSURE

• A manometer is a U-shaped tube that is partially filled with liquid.

• Both ends of the tube are open to the atmosphere.

• A container of gas is connected to one end of the U-tube.

• If there is a pressure difference between the gas and the atmosphere, a force will be exerted on the fluid in the U-tube. This changes the equilibrium position of the fluid in the tube.

MEASURING PRESSURE

'

atm

gauge

B B C

B C B

let d h

P P P h D g

P P P P h D g

P h D g

The pressure at point B is the pressure of the gas.

atmc PP

B'B PP

MEASURING PRESSURE

A Barometer The atmosphere pushes on the container of mercury which forces mercury up the closed, inverted tube. The distance d is called the barometric pressure.

atmBA PPP

A

let d h

P h D g

Atmospheric pressure is equivalent to a column of mercury 76.0 cm tall.

Absolute pressure: The pressure of a fluid is expressed relative to that of vacuum (=0)

Gage pressure: Pressure expressed as the difference between the pressure of the fluid and that of the surrounding atmosphere.

Usual pressure gages record gage pressure. To calculate absolute pressure:

ABSOLUTE AND GAGE PRESSURE

gageatmabs PPP

UNITS FOR PRESSURE

Unit Definition or Relationship

1 pascal (Pa) 1 kg m-1 s-2

1 bar 1 x 105 Pa

1 atmosphere (atm) 101,325 Pa

1 torr 1 / 760 atm

760 mm Hg 1 atm

14.696 pounds per sq. in. (psi)

1 atm

Pascal’s 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.

PASCAL'S LAW

DIRECTION OF FLUID PRESSURE ON BOUNDARIES

Furnace duct Pipe or tube

Heat exchanger

Dam

Pressure is a Normal Force(acts perpendicular to surfaces)It is also called a Surface Force

At any point within a liquid, the forces that produce pressure are exerted equally in all directions.

Pressure increases vertically downward.

Pressure is constant horizontally.

FLUID PRESSURE

When the liquid is pressing against a surface there is a net force directed perpendicular to the surface.

If there is a hole in the surface, the liquid initially moves perpendicular to the surface.

At greater depths, the net force is greater and the horizontal velocity of the escaping liquid is greater

FORCES EXERTED BY A FLUID

Pascal’s Principle: A CHANGE IN PRESSURE IN A CONFINED FLUID IS TRANSMITTED WITHOUT CHANGE TO ALL POINTS IN THE FLUID.

Ex. Hydraulic lift. Hydraulic piston apparatus

uses an incompressible fluid to transmit pressure from a small cylinder to a large cylinder.

According to Pascal’s Principle, the pressure in the small cylinder resulting from the application of F1 to a frictionless piston is transmitted undiminished to the larger piston.

TRANSMISSION OF PRESSURE: PASCAL’S PRINCIPLE.

APPLICATION OF PASCAL'S LAW

Apply a force F1 here to a piston of cross-sectional area A1.

The applied force is transmitted to the piston of cross-sectional area A2 here.

F2 F1A2A1

Pressure of a liquid does not depend on the amount of liquid.

HYDROSTATIC PARADOX

At a given depth, a given liquid exerts the same pressure against any surface - the bottom or sides of its container, or even the surface of an object submerged in the liquid to that depth.

Pressure a liquid exerts depends only on its density and depth

COLUMNAR FLUID PRESSURE

( )absolute oP P h D g peanut hot dog

PRESSURE MEASUREMENT

Types of MeasurementMechanical

U-tube manometer, Bourdon tube, Diaphragm and Bellows

Electrical Strain Gauge, Capacitive sensor, Potentiometric,

Resonant Wire, Piezoelectric, Magnetic, Optical

MECHANICAL

Mechanical pressure measurement devices are large and cumbersome.

Not suited for automated control loops typical in industry.

Mechanical devices:U-tube ManometerBourdon tubeDiaphragm and Bellows element

U-TUBE MANOMETER

Measures difference in pressure between two points in a pipe.

Typical in laboratories.

BOURDON TYPE

Flexible element used as sensor.Pressure changes cause change in

element position.Element connected to pointer to

reference pressure.

DIAPHRAGM AND BELLOWS ELEMENT

Similar concept to Bourdon type.Widely used because they require less

space and can be made from materials that resist corrosion.

STRAIN GAUGE

Measures deflection of elastic diaphragm due to pressure difference across diaphragm.

Widely used in industry.Used for small pressure ranges.Measurements tend to drift.

CAPACITIVE SENSOR

Measures changes in capacitance of electrically charged electrodes from movement of metal diaphragm due to pressure difference across diaphragm.

CAPACITIVE SENSOR, CONT.

Can be operated in balanced or unbalanced mode. Balanced always has capacitance of zero. Measures

pressure indirectly by measuring drift in capacitor arms. Unbalanced measures ratio between output voltage and

excitation voltage.Widely used in industry.Large rangeability.

PIEZOELECTRIC

Measures the charge developed across quartz crystal due to change in pressure.

Charge decays rapidly making unsuitable for static pressure measurements.

Sensors are very rugged. Pressure can be applied longitudinally or transversally.

Used to measure dynamic pressure changes associated with explosions and pulsations .

MAGNETIC

Measures induced current caused by movement of magnetic components from pressure changes.

Used in applications where high resolution in small range is desired due to very high output signals.

Sensitive to stray magnetic fields and temperature changes.

Various types of pressure gauges

PRESSURE MEASUREMENT

U-tube Manometer

Principles: Hydrostatic Law

∆P=ρ g h

http://www.efunda.com/formulae/fluids/images/Manometer_A.gif

HYDROSTATIC FORCES ON PLANE SURFACES

On a plane surface, the hydrostatic forces form a system of parallel forces

For many applications, magnitude and location of application, which is called center of pressure, must be determined.

Atmospheric pressure Patm can be neglected when it acts on both sides of the surface.

PRESSURE PRISM

1

2

Average pressure on the surface Pavg= γ(h1+h2)/2

Resultant force on the surface FR= γ(h1+h2)A/2

Distance of point of action of force from the surface

Yc= Y coordinate of centroid

hc= distance of centroid from the surface

,xx Cp C

c

Iy y

y A

RESULTANT FORCE

The magnitude of FR acting on a plane surface of a completely submerged plate in a homogenous fluid is equal to the product of the pressure PC at the centroid of the surface and the area A of the surface

Consider the top surface of a flat plate of arbitrary shape completely sub-merged in a liquid, as shown in Fig. together with its top view.

absolute pressure at any point on the plate is

The resultant hydrostatic force FR acting on the surface is determined by integrating the force P dA acting on a diff erential area dA over the entire surface area,

CENTER OF PRESSURE

Line of action of resultant force FR=PCA does not pass through the centroid of the surface. In general, it lies underneath where the pressure is higher.

Vertical location of Center of Pressure is determined by equation the moment of the resultant force to the moment of the distributed pressure force.

$Ixx,C is tabulated for simple geometries.

,xx Cp C

c

Iy y

y A

HYDROSTATIC FORCES ON CURVED SURFACES

FR on a curved surface is more involved since it requires integration of the pressure forces that change direction along the surface.

Easiest approach: determine horizontal and vertical components FH and FV separately.

STABILITY OF IMMERSED BODIES

Rotational stability of immersed bodies depends upon relative location of center of gravity G and center of buoyancy B. G below B: stable G above B: unstable G coincides with B: neutrally stable.

STABILITY OF FLOATING BODIES

If body is bottom heavy (G lower than B), it is always stable.

Floating bodies can be stable when G is higher than B due to shift in location of center buoyancy and creation of restoring moment.

Measure of stability is the metacentric height GM. If GM>1, ship is stable.