cbe 150a – transport spring semester 2014 fluid statics

CBE 150A – Transport Spring Semester 2014

Fluid Statics


Definition of Pressure

Pam

N

area

forceP

2

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


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

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

Usual pressure guages record guage pressure. To calculate Usual pressure guages record guage pressure. To calculate absolute pressure: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 Let’s determine the pressure distribution in a fluid at rest distribution in a fluid at rest in which the only body force in which the only body force acting is due to gravityacting is due to gravity

The sum of the forces acting The sum of the forces acting on the fluid must equal zeroon the fluid must equal zero


mg

zgSz

PS

x

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?

zzPS


Pressure distribution for a fluid at rest

zgSPSPSFzzzz

0

gz

PP zzz

gdz

dP

A force balance in the z direction gives:

For an infinitesimal element (z0)


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


Example

In deep water oil fields offshore Louisiana one occasionally encounters a reservoir pressure of 10,000 psig at a depth of 15,000 ft. If this pressure is greater than the hydrostatic pressure of the drilling fluid in the well bore the result may be a well “blowout” which is dangerous to life and property. Assuming that you are responsible for selecting the drilling fluid for an area where such pressures are expected, what is the minimum density drilling fluid you can use ? Assume a surface pressure of 0 psig.


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 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


Practical Application of Buoyancy

Nimitz Class Aircraft Carrier (100,000 tons)


Measurement of Pressure

Manometers are devices in which one or more Manometers are devices in which one or more columns of a liquid are used to determine the columns of a liquid are used to determine the pressure difference between two points.pressure difference between two points.

– U-tube manometerU-tube manometer

– Inclined-tube manometerInclined-tube manometer


Measurement of Pressure Differences

mambb

mmba

gRZgPP

RZgPP

)(

)(

3

2

)( bamba g RPP

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


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 Gases are compressible i.e. their density varies with

temperature and pressure temperature and pressure ==P M /RTP M /RT

– For small elevation changes (as in engineering For small elevation changes (as in engineering applications, tanks, pipes etc) we can neglect the applications, tanks, pipes etc) we can neglect the effect of elevation on pressureeffect of elevation on pressure

– In the general case start from: In the general case start from:

gdz

dP

o

o

RT

zzMgPP

Tfor

)(exp

:constT

1212


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 linearAssume 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.

cbe 150a – transport spring semester 2014 fluid statics

Documents

surface pressure

pressure distribution

direction of fluid pressure

pressure unitdefinition

hydrostatic pressure

reservoir pressure

transport spring semester

p z z