Fluid Statics

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

Pam

N

area

forceP

2

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

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 guages record guage pressure. To calculate absolute pressure:

gaugeatmabs 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

Pressure distribution for a fluid at rest

Let’s 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

mg zgSx

y

zLet Pz and Pz+z denote the pressures at the base and top of the cube, where the elevations are z and z+z respectively.

What are the z-direction forces?

zPS

zzPS

Pressure distribution for a fluid at rest

zgSPSPSFzzzz

0

gz

PP zzz

A force balance in the z direction gives:

For an infinitesimal element (z0)

gdz

dP

Incompressible fluidLiquids are incompressible i.e. their density is assumed to be constant:

By using gauge pressures we can simply write:

Po is the pressure at the free surface (Po=Patm)

When we have a liquid with a free surface the pressure P at any depth below the free surface is:

)zz(gPP 1212

PghP o

ghP

Static pressure is a concern !!!!!

Static pressure - 10 minute problem:

Rainwater collects behind the concrete retaining wall shown below. If the water-saturated soil (specific gravity = 2.2) acts as a fluid, determine the force and center of pressure on a one-meter width of wall.

Water

Soil 3m

1m

Buoyancy• 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 floatsFree liquid surface

The upper surface of the body is subjected to a smaller force than the lower surface

A net force is acting upwards

F1

F2

h1

h2

H

BuoyancyThe net force due to pressure in the vertical direction is:

FB = F2- F1 = (Pbottom - Ptop) (xy)

The pressure difference is:

Pbottom – Ptop = g (h2-h1) = g H

Combining:

FB = g H (xy)

Thus the buoyant force is:

FB = g V

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

Measurement of Pressure Differences

mambb

mmba

gRZgPP

RZgPP

)(

)(

3

2

Apply the basic equation of static fluids to both legs of manometer, realizing that P2=P3.

)( bamba gRPP

Inclined Manometer

• To measure small pressure differences need to magnify Rm some way.

sin)(1 baba gRPP

Manometer - 10 minute problem

A simple U-tube manometer is installed across an orifice plate. The manometer is filled with mercury (specific gravity = 13.6) and the liquid above the mercury is water. If the pressure difference across the orifice is 24 psi, what is the height difference (reading) on the manometer in inches of mercury ?

Compressible Flow:

Natural gas well

Tall Mountains

Compressible fluid• Gases are compressible i.e. their density varies with

temperature and pressure =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:

o

o

RT

zzMgPP

Tfor

)(exp

:constT

1212

gdz

dP

CompressibleLinear Temperature Gradient

)( 00 zzTT

z

z

p

p zzT

dz

R

Mg

p

dp

00)( 00

RMg

T

zzTpzp

0

000

)()(

Atmospheric Equations

• Assume linear

RMg

T

zzTpzp

0

000

)()(

• Assume constant

0

0 )(

0)( RTzzMg

epzp

Temperature variation with altitude for the U.S. standard atmosphere

Compressible Isentropic

v

p

CC

Pconstant

P

1

1y

P

P

T

T1

11

112

1

112

11

11

RT

zgMTT

RT

zgMPP

Compressible Fluid – 10 minute problem

The temperature of the earth’s surface drops about 5 C for every 1000 m of elevation above the earth’s surface. If the air temperature at ground level is 15 C and the pressure is 760 mm Hg, what is the air pressure on top of Mt. Everest at 8847 m ? Assume air behaves as an ideal gas.