Fundamentals of Fluid Mechanics, 5/E Lecture 2 Fluid Statics and Pressure Measurements Unit...

Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi

Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Lecture 2

Fluid Statics and Pressure Measurements

Unit Operations 2 CHE3012



Overview

•Pressure at a point

•Basic Equation for Pressure Field

•Pressure variation in a fluid at rest

•Standard Atmosphere

•Measurement of Pressure and Pressure

Measuring Devices




Figure 2.1 (p. 39) Forces on an arbitrary wedge-shaped element of fluid.

Pressure at a Point

Free-body Diagram of a Fluid wedge




Assumptions

•zero shearing stress (fluid element moves as a rigid body)

•the only forces are due to pressure and weight

•Neglect forces in the x-direction (for simplicity)

ysyya

zyxsxpzxpF

maF

2sin

From Newton’s 2nd Law of motion

zszz azyxzyx

sxpyxpF22

cos

sin,cos szsy

0,,limit

2

2

zyxthetake

zapp

yapp

zsz

ysy

szsy pppp ,

Conclusion

Pressure at a point in a fluid at rest or in motion, is independent of direction as

long as there are no shearing stress. Pascal’s Law




Figure 2.2 (p. 40) Surface and body forces acting on

small fluid element.

Basic Equation for Pressure Field

Assumptions

•zero shearing stress (fluid element

moves as a rigid body)

•only surface force due to pressure and

body force equal to the weight of the

element are acting on the element

•Neglect other types of body forces

•Neglect forces in the x-direction (for

simplicity)




in the y-direction

zyxy

pF

zxy

y

ppzx

y

y

ppF

y

y

22

zyxz

pF

zyxx

pF

z

x

in the x- and z-directions

zyxkz

pj

y

pi

x

pF

or

kFjFiFF

s

zyxs

ˆˆˆ

ˆˆˆ

pkz

pj

y

pi

x

p

ˆˆˆpressure gradient

Resultant surface force




kz

jy

ix

ˆˆˆ

where

thus, the resultant surface force pzyx

Fs

the weight in the z-direction

kzyxkW ˆˆ

akp

therefore

azyxkzyxzyxp

or

amkWFF s

ˆ

ˆ

ˆ




For a fluid at rest a = 0,

0

x

p

0ˆ kp

z

p0

y

p

Thus as we move from point to point on the horizontal plane the pressure does not

change.

dz

dp

Pressure Variation in a Fluid at a Rest Unit Operations 2 CHE3012



Figure 2.3 (p. 43)

Notation for pressure variation in a fluid at

rest with a free surface.

dz

dp

1212

2

1

2

1

zzppdzdpp

p

z

z

Incompressible Fluid (Hydrostatic Pressure)

1221 zzpp

hpp 21

21 pp

h

(Pressure head)

ophp

P2 represent your datum P0




Figure 2.4 (p. 44) Fluid equilibrium in a container of arbitrary shape.




Figure E2.1

Question 1

Because of a leak in a buried

gasoline storage tank, water

has seeped in to the depth

shown in the figure. If the

specific gravity of gasoline is

SG = 0.68, determine the

pressure at the gasoline water

interface and at the bottom of

the tank. Express the answer

as absolute pressure, gauge

pressures and pressure head.

m

m




Figure 2.5 Transmission of fluid pressure.

The pressure in a closed system is transmitted equally throughout.

1

212 A

AFF




dz

dp

RTp

Compressible Fluid

RT

gp

dz

dp

Gases are considered compressible fluids, since the density can change

considerably with changes in pressure and temperature.

Specific weight for gases are relatively small when compared to liquids, thus it

follows that from the equation the pressure gradient in the vertical direction is

correspondingly small, and even over distance of several hundred feet the

pressure will remain essentially constant for a gas.

For large variation in height we recall:

These relationships can be combined to give




Compressible Fluid

2

1

2

1

z

z

p

p T

dz

R

g

p

dp

By separating the variables we get

To complete this integration the relationship between temperature and

elevation must be specified. If we assume constant temperature T0

over the range z1 to z2 (isothermal condition), then

0

1212 exp

RT

zzgpp




Table 2.1 (p. 47) Properties of U.S. Standard Atmosphere at Sea Level

Standard Atmosphere Unit Operations 2 CHE3012



Figure 2.6 (p. 47) Variation of temperature with altitude in the U.S. standard atmosphere.



Measurement of Pressure and Pressure

Measuring Devices

Monometers

Mechanical and Electronic Pressure measuring devices




0php 02211 hhpA

Piezometer Tube

Fundamental equation describing their use

U-Tube Manometer

1122 hhpA 11hpA

22hpA

If fluid A was a gas then



02211 hhpA

113322BA

B332211A

hγhγhγpp

phγhγhγp

Differential U-tube manometer. Incline-Tube manometer.

113322BA

B332211A

hγhγsinθlγpp

phγsinθlγhγp

sinθγ

ppl

or

sinθlγpp

2

BA2

22BA

If pipes A and B contain a gas



Questions???


Fundamentals of Fluid Mechanics, 5/E Lecture 2 Fluid Statics and Pressure Measurements Unit...

Documents

Transcript of Fundamentals of Fluid Mechanics, 5/E Lecture 2 Fluid Statics and Pressure Measurements Unit...