Fluid Mechanics

FLUID MECHANICS

Manila Review Institute Chemical Engineering Review

junolano@yahoo.com

Dr. Servillano Olano, Jr

Fluid Mechanics

� Fluid mechanics

� Branch of engineering science that has to do with the behavior of fluids (liquids, gases and vapors)

� Branches of Fluid Mechanics

1. Fluid Statics

� Fluids in equilibrium state of no shear stress

2. Fluid Dynamics

� Portions of the fluid are in motion relative to

the other parts

Fluid Statics

Nature of Fluids

A fluid is a substance that does not

permanently resist distortion.

Some physical properties of fluids:

a) Density or relative density

b) Viscosity

c) Surface tension

� Types of Fluids

1. Incompressible

� Density is not affected by changes in temperature and pressure

2. Compressible

� Density varies appreciably with temperature and pressure

� Pressure Concept

� For a static fluid, the pressure at any point is independent of direction.

Fluid Mechanics

English SI

Length, L ft., inch meter (m)

Mass, M lbm, slugs Kg

Time, T seconds (s), hr s

Force, F lbf newton (N)

Density, ρ lbm/ft3, slug/ft3 Kg/m3

Systems of Units

Relationship between Force and Mass: F=ma

2

2

32.16ENGLISH:

32.16c

ftg lbfsF m lbm lbm lbf

lbm ftg lbm

lbf s

= = =⋅⋅

2 2SI: 9.806 ( )m mF mg kg kg newton Ns s

= ⋅ = ⋅ =

Fluid Statics and Applications

Hydrostatic Equilibrium

Force balance:

- ( ) - 0

0

c

c

gp S p dp S Sdz

g

gdp dz

g

ρ

ρ

+ =

+ =

Fluid Statics and Applications

2 11 2

constant

( )

c

c

p gz

g

p p gz z

g

ρ

ρ ρ

+ =

− = −

Hydrostatic Equilibrium,

for constant density (most liquids)

0

0

c

c

pM gdp dz

RT g

dp gMdz

p g RT

+ =

+ =

Barometric EquationFor an ideal gas, ρ = (pM/RT).

Substituting,

Fluid Statics and Applications

Integrating between levels 1 and 2:

( )

( )

22 1

1

2 12

1

ln

exp

c

c

p gMz z

p g R T

gM z zp

p g R T

= − −

− = −

(Called the barometric equation)

Fluid Statics and Applications

2. Simple Manometers

Pressure balance at level 0:

ρρρ

ρρρρρ

zg

gH

g

gpp

g

gH

g

ga

g

gzp

g

ga

g

gHp

c

mm

c

c

mm

cccc

m

∆+−=−

++∆+=++

)(21

21

Simplifying gives:

Fluid Statics and Applications

Two-fluid U-tube Manometer

Pressure balance at point 0:

=∴

=

−+−=−

++=++

A

aHh

hAaHbut

ghgHpp

gHhpgHhp

m

m

ABBmm

mmBBmA

)()(

)()(

21

21

ρρρρ

ρρρρ

Viscous forces in a fluid

Rheological Properties of Fluids

Evaluation of Fanning friction factor

Fluid Dynamics and Applications

Evaluation of surface roughness factor

Flow of Incompressible Fluids

3. Turbulent flow in pipes and closed channels

(correlation equations)

(Values of α and β are close to unity)

(((( ))))

(((( )))) 400041

6000641

8

51

915

81

2

2

.fNlog.f

.fNlog.f

fk

fkk

f

Re

Re

−−−−====

−−−−====

++++====

−−−−++++====

ββββ

ααααk = 0.40

Von Karman eq

Nikuradze eq.

Flow of Incompressible Fluids

Other Correlations for f:

Blassius Formula: (for smooth tubes)

Colebrook Equation:

Churchill Equation:

Evaluation of Ff (for fittings and valves)

Entrance section of a pipe

� For fully developed velocity profile:

For laminar flow:

For turbulent flow:

Ree N.

D

L05750====

50≅≅≅≅D

Le

Le

Coverage Chart

Classification of Pumps

Examples of Pumps

A. Centrifugal pumps

Dynamic pumps

B. Reciprocal pumps

Positive displacement pumps

C. Gear pumps

D. Axial flow pumps

Simple Centrifugal Pump

Examples of Pumps

Examples of Pumps

Examples of Pumps

Characteristic Curves

Characteristic Curves of Centrifugal Pumps

Guide in the selection of Pumps

System head vs Available head

Net Positive Suction Head (NPSH)

� Head available at the pump inlet to keep the liquid from cavitating or boiling

Where: ps = pressure at suction point

pv = vapor pressure of the liquid

If NPSH <= 0, cavitation will occur

Note: NPSH(available) should be greater than NPSH(required)

ρρ

ρρ

v

c

s

c

a

vsa

pF

g

v

g

gz

pNPSH

ppNPSH

−∑−−+=

−=

)2

(

2

11

Flow Meters

a) U-Tube Manometer

p1 – p2 = Hm (ρm - ρ)

or

where: Hm = manometer reading

∆H = differential head

ρm = density of manometer liquid

−−−−

ρρρρ

ρρρρ====

ρρρρ

−−−−====∆∆∆∆ 1H

ppH m

m21

Pitot Tube

If tube opening is placed at the center,

(for incompressible fluids)

where

vcpmax Hg2Cv ∆∆∆∆====

−−−−

ρρρρ

ρρρρ====∆∆∆∆ 1HH m

mv

Flow Metersa) Pitot Tube (measurement of local velocity)

By MEB Eq:

ρρρρ

−−−−==== c12

p

g)pp(2Cv

Pitot TubeTo get average velocity, vav:

Where

see Fig. 2.10-2 G, to get vav

For gases at velocities > 200 fps, see Eq. 10-8, Perry.

(((( ))))maxRe,Re

max

av NorNfv

v====

µµµµ

ρρρρ==== max

maxRe,

DvN

Flow Metersc) Head Meters (Orifices, venturi meters,

nozzles)

Head MetersEvaluation of Y: f (type of fluid)

�For liquids, Y = 1.0

�For gases, see Fig. 10-16, Perry

Evaluation of C (discharge coefficient)

Types of Taps:

Rotameters (or Area Meters)

� Force Balance:

where

� vf = velocity of float

� ρf = density of float

� Af = max. cross-sectional area of float

)p(Ag

g)(v f

c

ff ∆∆∆∆−−−−====ρρρρ−−−−ρρρρ

cf

ff

gA

g)(vp

ρρρρ−−−−ρρρρ====∆∆∆∆

Rotameters

� In most cases, the geometry of the rotameter is not known, so a calibration curve using water is prepared. To determine flowrates for other liquids or gases, the above relation is used.

� In terms of velocity

� For an identical flowmeter

(a linear relationship)

f

ffoR111

A

v)(g2ACAv

ρρρρ−−−−ρρρρρρρρ====ρρρρ

o

'

R1 ACv ≈≈≈≈

Rotameters (or Area Meters)

� Substituting in the General Equation:

� Since 1 - β4 ≅ 1.0

� For values of CR, see textbook or other references

)1(A

v)(g2ACm

4

f

ffoR

ββββ−−−−

ρρρρ−−−−ρρρρρρρρ====

f

ffoR

A

v)(g2ACm

ρρρρ−−−−ρρρρρρρρ====

Flow in Open Channels and Weirs

1. Rectangular Weir

(Modified Francis Weir Formula)

)g2h)(h2.0L(415.0q 5.1

oo−−−−====

φφφφ====

tan

g2h31.0q

5.2

o

2. Triangular Weir Notch

Note: Both equations apply only for water

Discussion of Problems

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