Equ

atio

ns S

umm

ary

from

Insi

de C

over

of M

orri

son,

201

3

Mec

hani

cal

Ener

gy B

alan

ce

Fann

ing

Fric

tion

Fact

or(p

ipe

flow

)

Dra

g C

oeff

icie

nt

(sph

ere

drop

)

Mom

entu

m b

alan

ce o

n a

CV

(R

eyno

lds t

rans

port

theo

rem

)

Mac

rosc

opic

Mom

entu

m B

alan

ce o

n a

CV

Nav

ier-

Stok

es e

quat

ion

(mic

rosc

opic

mom

entu

m b

alan

ce,

inco

mpr

essi

ble,

New

toni

an fl

uids

)

Con

tinui

ty e

quat

ion

(mic

rosc

opic

mas

s bal

ance

, in

com

pres

sibl

e flu

ids)

Hyd

rost

atic

Pre

ssur

e

Hag

en-P

oise

uille

Equ

atio

n (s

tead

y, la

min

ar tu

be fl

ow,

inco

mpr

essi

ble)

Pran

dtl E

quat

ion

(ste

ady,

turb

ulen

t tub

e flo

w)

Stok

es-E

inst

ein-

Suth

erla

nd E

quat

ion

(ste

ady,

slow

flow

aro

und

a sp

here

)

Notethisiscorrect;thereisan

erroro

ntheinside

cover

4c ©

2011FaithA.Morrison,allrights

reserved.

Total

stress

tensor

Π̃=

−pI

+τ̃

⎛ ⎝Π̃11

Π̃12

Π̃13

Π̃21

Π̃22

Π̃23

Π̃31

Π̃32

Π̃33

⎞ ⎠ 123

=

⎛ ⎝τ̃11−

pτ̃ 1

2τ̃ 1

3

τ̃ 21

τ̃ 22−p

τ̃ 23

τ̃ 31

τ̃ 32

τ̃ 33−p

⎞ ⎠ 123

Dynam

icpressure

P≡

p+ρgh

New

tonian

constitutive

equation

τ̃=

μ( ∇v

+(∇

v)T)

=μ

⎛ ⎜ ⎜ ⎝2∂v1

∂x1

∂v2

∂x1+

∂v1

∂x2

∂v3

∂x1+

∂v1

∂x3

∂v2

∂x1+

∂v1

∂x2

2∂v2

∂x2

∂v2

∂x3+

∂v3

∂x2

∂v3

∂x1+

∂v1

∂x3

∂v2

∂x3+

∂v3

∂x2

2∂v3

∂x3

⎞ ⎟ ⎟ ⎠ 123

Total

molecularfluid

force

onafinitesurfaceS

F=

∫∫ S[ n̂·

Π̃] ats

urface

dS

Station

aryfluid

[ n̂·Π̃] =

−pn̂

Movingfluid

[ n̂·Π̃] =

−pn̂

+n̂·τ̃

Total

fluid

torque

onafinitesurfaceS

T=

∫∫ S[ R ×

( n̂·Π̃)] at

surface

dS

Total

flow

rate

out

through

afinitesurfaceS

Q=

V̇=

∫∫ S[n̂

·v] atsu

rface

dS

Average

velocity

across

afinitesurfaceS

〈v〉=

Q S

c ©2011FaithA.Morrison,allrights

reserved.

5

Coordinatesystem

surfacedifferential

dS

Cartesian

(top

,n̂=

ê z)

dS=

dxdy

Cartesian

(sidea,

n̂=

ê y)

dS=

dxdz

Cartesian

(sideb,n̂=

ê x)

dS=

dydz

cylindrical(top

,n̂=

ê z)

dS=

rdrdθ

cylindrical(side,

n̂=

ê r)

dS=

Rdθdz

spherical,(n̂

=ê r)

dS=

R2sinθdθdφ

Coordinatesystem

volumedifferential

dV

Cartesian

dV

=dxdydz

cylindrical

dV

=rdrdθdz

spherical

dV

=r2

sinθdrdθdφ

Coordinatesystem

coordinates

basisvectors

spherical

x=

rsinθcosφ

ê r=

(sinθcosφê x)+(sinθsinφê y)+cosθê

z

y=

rsinθsinφ

ê θ=

(cos

θcosφ)ê

x+(cos

θsinφ)ê

y+(−

sinθ)ê z

z=

rcosθ

ê φ=

(−sinφ)ê

x+cosφê y

cylindrical

x=

rcosθ

ê r=

cosθê

x+sinθê

y

y=

rsinθ

ê θ=

(−sinθ)ê x

+cosθê

y

z=

zê z

=ê z

Divergence

Theorem

∫∫ Sn̂·F

dS

=

∫∫∫V∇

·FdV

StokesTheorem

∮ Ct̂·F

dl

=

∫∫ Sn̂·(∇

×F)dS

Vectoridentities:

∇·∇

×F

=0

(Divergence

ofcurl=

0)

∇×

∇f

=0

(Curlof

grad

ient=

0)

∇(fg)

=f∇g+g∇f

F·∇F

=1 2∇

( F2) −

F×(∇

×F)

∇·(fF)

=f∇

·F+F·∇f

∇×

∇×

F=

∇(∇

·F)−∇

2F

∇·(F

×G)

=G

·(∇

×F)−F·(∇

×G)

Theeq

uatio

nsinF.A.

Morrison

,AnIntrod

uctio

nto

FluidMecha

nics

(Cam

bridge,2013)

assumethefollowingde

finition

softhe

cylindrical

andsphe

ricalcoordinate

system

s .

Cylin

dricalCo

ordina

teSystem

:Notethat

the

coordinate

swings

arou

ndthe

axis

Sphe

ricalCo

ordina

teSystem

:Notethat

the

coordinate

swings

downfrom

the

axis;

thisisdiffe

rent

from

itsde

finition

inthecylindricalsystem

above.

Ver.21

Sep20

11

FACT

ORS

FORUNITCO

NVER

SIONS

Qua

ntity

Equivalent

Value

s

Mass

1kg

=10

00g=0.00

1metricton=2.20

462lb

m=35

.273

92oz

1lb

m=16

oz=5x10

4ton=45

3.59

3g=0.45

3593

kg

Length

1m

=10

0cm

=10

00mm

=10

6microns

(μm)=

1010angstrom

s(Å)

=39

.37in=3.28

08ft=1.09

36yd

=0.00

0621

4mile

1ft=12

in.=

1/3yd

=0.30

48m

=30

.48cm

Volume

1m

3=10

00liters=10

6cm

3=10

6ml

=35

.314

5ft3=22

0.83

impe

rialgallons

=26

4.17

gal

=10

56.68qt

1ft3=17

28in

3=7.48

05gal=

0.02

8317

m3=28

.317

liters

=28

317cm

3

Force

1N=1kg

. m/s

2=10

5dyne

s=10

5g.cm

/s2=0.22

481lb

f

1lb

f=32

.174

lbm. ft/s

2=4.44

82N=4.44

82x10

5dyne

s

Pressure

1atm

=1.01

325x10

5N/m

2(Pa)=10

1.32

5kPa=1.01

325bars

=1.01

325x10

6dyne

s/cm

2

=76

0mm

Hgat

0C(torr)=10

.333

mH2O

at4

C=14

.696

lbf/in

2(psi)=

33.9ftH2O

at4C

100kPa=1bar

Energy

1J=

1N. m

=10

7ergs

=10

7dyne

. cm

=2.77

8x10

7kW

. h=0.23

901cal

=0.73

76ft. lb

f=9.47

817x10

4Btu

Power

1W

=1J/s=0.23

885cal/s=0.73

76ft. lb

f/s=9.47

817x10

4Btu/s=3.41

21Btu/h

=1.34

1x10

3hp

(horsepo

wer)

Viscosity

1Pa

. s=1N. s/m

2=1kg/m

. s=10

poise=10

dyne

s. s/cm

2=10

g/cm

. s=10

3cp

(cen

tipoise)

=0.67

197lb

m/ft. s

=24

19.088

lbm/ft. h

Den

sity

1kg/m

3=10

3g/cm

3

=0.06

243lb

m/ft3

103kg/m

3=1g/cm

3=62

.428

lbm/ft3

VolumetricFlow

1m

3 /s=

35.314

5ft3 /s=15

,850

.2gal/min(gpm

)1gpm

=6.30

907x10

5m

3 /s=2.22

802x10

3ft3 /s=3.78

54liter/m

in1liter/m

in=0.264

17gpm

Prof.Faith

A.M

orrison

Dep

artm

ento

fChe

micalEn

gine

ering

Ver.21

Sep20

11

Tempe

rature

5 9(

)(

)32

oo

TC

TF

9 5(

)(

)32

1.8

()

32o

oo

TF

TC

TC

AbsoluteTempe

rature

T(K)

=T(

C)+27

3.15

T(R)

=T(

F)+45

9.67

Tempe

rature

Interval(T)

1C

=1K=1.8F

=1.8R

1F=1R

=(5/9)C

=(5/9)K

USEFU

LQUANTITIES

SG=

(20C)/

water(4

C)

water(4

C)=1000

kg/m

3=62

.43lb

m/ft3

=1.00

0g/cm

3

water(25C)

=997.08

kg/m

3=62.25lb

m/ft3=0.99709g/cm

3

g=9.8066

m/s

2=98

0.66

cm/s

2=32

.174

ft/s

2

μ water(25C)

=8.937x10

4Pa

. s=8.93

7x10

4kg/m

. s=0.8937

cp=0.8937x10

2g/cm

. s=6.005x10

4lb

m/ft. s

Compo

sitio

nof

air:

N2

78.03%

O2

20.99%

Ar

0.94

%CO

20.03

%H2,He,Ne,Kr,Xe

0.01

%100.00%

Mair

=29

g/mol=29

kg/kmol=29

lbm/lbm

ole

p,water(25C)

=4.182kJ/kgK=0.99

89cal/gC=0.9997

Btu/lb

mF

R=8.314m

3 .Pa/m

ol. K=0.08314liter

. bar/m

ol. K=0.08206liter

. atm

/mol

. K=62

.36liter

. mm

Hg/mol

. K=0.7302

ft3 .atm/lbm

ole.R

=10

.73ft3 .psia/lbm

ole.R

=8.314J/mol

. K=1.987cal/mol

. K=1.987Btu/lbmole.R

1CM

3110

Morrison

Mich

igan

Tech

2020

FluidFrictionDa

taCo

rrelations

forE

xaminations

CM31

10TransportP

heno

men

aI

Mich

igan

Techno

logicalU

niversity

ProfessorF

aith

A.Morrison

1De

cembe

r202

0

I. DataCorrelationsforFlow

throughSm

oothPipes

A. AllReynoldsnumbers:Morrison

Thecorrelationfro

mMorrison

(201

3)fitsthe

smoo

thpipe

data

fora

llRe

ynolds

numbe

rs;b

eyon

dRe

4000thiscorrelationfollowsthe

Prandtlequ

ation(see

Figure

1;Morrison

,equ

ation7.15

8).This

correlationisexplicitin

;whe

nflo

wrate

isknow

n,may

befoun

ddirectly;w

hen

isknow

n,or

mustb

esolved

foriterativ

ely.

Morrison

(201

3)0.0076

3170 Re.

13170 Re.

16 Re(1)

B.

,RRe

:Prandtl

ThePrandtlcorrelatio

nfor

Reinturbu

lent

flowisno

texplicitinfrictionfactor

andmay

besolved

iterativ

elyexcept

whe

nisknow

n(M

orrison

,equ

ation7.15

6).Whe

nvelocityisno

tkno

wn,

Re may

stillbe

calcu

lated,yielding

.ThePrandtlcorrelatio

nisgood

onlyforRe

4,000.Prandtl

orVo

nKarman

Nikuradse

(Den

n,19

80)

1 4.0logR

e0.40

(2)

C.

,Re

:Asimplified

Correlation

Forthe

turbulen

tregim

e,an

approxim

atecorrelationthat

ismuchsim

pler

toworkwith

(with

acalcu

latoro

nan

exam

,for

exam

ple)

isgivenhe

reandshow

ninFigure

2(M

orrison

,equ

ation7.15

7).

Thisisgood

onlyforRe

4,000.Simplified

Turbulen

t(W

hite,197

4)1.02 4lo

gRe.

(3)

2CM

3110

Morrison

Mich

igan

Tech

2020

Figure

1:Eq

uatio

n3captures

smoo

thpipe

frictionfactor

asafunctio

nof

Reynolds

numbe

roverthe

entireRe

ynolds

numbe

rrange

(Morrison

,201

3)andisrecommen

dedforspreadshe

etuse.Also

show

nareNikuradse'se

xperim

entaldataforflowinsm

ooth

pipe

s(Nikuradse,19

33).Us

ebe

yond

Re10

isno

trecom

men

ded;

forR

e>40

00eq

uatio

n3followsthe

Prandtlequ

ation.(M

orrison

,201

3,p5

32)

Figure

2:Forturbu

lent

flow,the

simplified

(equ

ation3)

orPrandtl(eq

uatio

n2)

correlations

may

beused

.Forw

orkwith

acalcu

lator,thesim

plified

correlationispe

rhapsthe

easie

stto

workwith

.(Morrison

,20

13,p53

1)

0.00

1

0.010.

11 1.E+

021.

E+03

1.E+

041.

E+05

1.E+

061.

E+07

Nik

urad

se (d

ata)

Lam

inar

flow

Pran

dtl (

Re>

4,00

0)

Sim

plifi

ed (R

e>4,

000)

Lam

inar

flow

Turb

ulen

t flo

w

f

Re

102

103

104

105

106

107

Smoo

th p

ipe

Pran

dtl C

orre

latio

n 40.0Re

log

0.41

ff

Re16f

Simplified

Correlation:

3CM

3110

Morrison

Mich

igan

Tech

2020

II.

DataCorrelationsforFlow

Around

aSphere

A. AllReynoldsNum

bers:M

orrison

Thecorrelationfro

mMorrison

(201

3)fitsthe

flowarou

ndasphe

refora

llRe

ynolds

numbe

rs(Figure3;

Morrison

equatio

n8.83

);be

yond

10thisc

orrelatio

nfollowsthe

curveshow

ninFigure

3.

Morrison

(201

3)24 Re

2.6Re 5.0 1Re 5.0.

0.411Re 263,00

0.1

Re 263,000.

0.25Re 10 1Re 10

(4)

Simplified

Correlations

Thecorrelations

below(M

orrison

,201

3;eq

uatio

n8.82

)are

simpler

relatio

nships

moresuita

bleto

calcu

lator/exam

work.

Re224 Re

(5)

0.1Re1,0

0024 Re1

0.14Re.

(6)

1,000Re2

.6100.445

(7)

2.810R

e10log Re 10

4.43logRe

27.3(8)

4CM

3110

Morrison

Mich

igan

Tech

2020

References

M.D

enn,ProcessF

luidMechanics

(Prentice

Hall,En

glew

oodCliffs,NJ,198

0)F.A.

Morrison

,AnIntrod

uctio

nto

FluidMecha

nics

(Cam

bridge

UniversityPress,Ne

wYo

rk,201

3).

F.M.W

hite,V

iscou

sFluidFlow

(McG

raw

Hill,Inc.:N

ewYo

rk,197

4).

Figure

3:Eq

uatio

n4captures

flowarou

ndasphe

reas

afunctio

nof

Reynolds

numbe

roverthe

entire

Reynolds

numbe

rrange

(Morrison

,201

3)andisrecommen

dedforspreadshe

etuse.Also

show

nare

expe

rimen

taldatafro

mWhite

(197

4).U

sebe

yond

10 isno

trecom

men

ded.(M

orrison

,201

3,p6

25)

TheEquation

ofContinuityand

theEquation

ofM

otion

inCartesian,

cylindrical,and

sphericalcoordinates

CM

3110Fall

2011Faith

A.M

orrison

ContinuityEquation,Cartesian

coordinates

∂ρ

∂t+

( v x∂ρ

∂x+vy∂ρ

∂y+

vz∂ρ

∂z

) +ρ

( ∂vx

∂x

+∂vy

∂y

+∂vz

∂z

)=

0

ContinuityEquation,cylindricalcoordinates

∂ρ

∂t+

1 r

∂(ρrvr)

∂r

+1 r

∂(ρvθ)

∂θ

+∂(ρvz)

∂z

=0

ContinuityEquation,spherical

coordinates

∂ρ

∂t+

1 r2∂(ρr2vr)

∂r

+1

rsinθ

∂(ρvθsinθ)

∂θ

+1

rsinθ

∂(ρvφ)

∂φ

=0

Equation

ofM

otionforan

incompressible

fluid,3compon

ents

inCartesian

coordinates

ρ

( ∂vx

∂t

+vx∂vx

∂x

+vy∂vx

∂y

+vz∂vx

∂z

)=

−∂P

∂x

+

( ∂τ̃xx

∂x

+∂τ̃ y

x

∂y

+∂τ̃ z

x

∂z

) +ρgx

ρ

( ∂vy

∂t+

vx∂vy

∂x

+vy∂vy

∂y

+vz∂vy

∂z

)=

−∂P ∂y+

( ∂τ̃xy

∂x

+∂τ̃ y

y

∂y

+∂τ̃ z

y

∂z

) +ρgy

ρ

( ∂vz

∂t+

vx∂vz

∂x

+vy∂vz

∂y

+vz∂vz

∂z

)=

−∂P ∂z+

( ∂τ̃xz

∂x

+∂τ̃ y

z

∂y

+∂τ̃ z

z

∂z

) +ρgz

Equation

ofM

otionforan

incompressible

fluid,3compon

ents

incylindricalcoordinates

ρ

( ∂vr

∂t+

vr∂vr

∂r

+vθ r

∂vr

∂θ

−v2 θ r+

vz∂vr

∂z

)=

−∂P ∂r+

( 1 r∂(rτ̃ r

r)

∂r

+1 r

∂τ̃ θ

r

∂θ

−τ̃ θ

θ r+

∂τ̃ z

r

∂z

) +ρgr

ρ

( ∂vθ

∂t+

vr∂vθ

∂r

+vθ r

∂vθ

∂θ

+vθvr

r+

vz∂vθ

∂z

)=

−1 r

∂P ∂θ+

( 1 r2∂(r

2τ̃ r

θ)

∂r

+1 r

∂τ̃ θ

θ

∂θ

+∂τ̃ z

θ

∂z

+τ̃ θ

r−

τ̃ rθ

r

) +ρgθ

ρ

( ∂vz

∂t+

vr∂vz

∂r

+vθ r

∂vz

∂θ

+vz∂vz

∂z

)=

−∂P ∂z+

( 1 r∂(rτ̃ r

z)

∂r

+1 r

∂τ̃ θ

z

∂θ

+∂τ̃ z

z

∂z

) +ρgz

Equation

ofM

otionforan

incompressible

fluid,3compon

ents

inspherical

coordinates

ρ

( ∂vr

∂t+

vr∂vr

∂r

+vθ r

∂vr

∂θ

+vφ

rsinθ

∂vr

∂φ

−v2 θ+v2 φ

r

)

=−∂P ∂r+

( 1 r2∂(r

2τ̃ r

r)

∂r

+1

rsinθ

∂(τ̃

θrsinθ)

∂θ

+1

rsinθ

∂τ̃ φ

r

∂φ

−τ̃ θ

θ+

τ̃ φφ

r

) +ρgr

ρ

( ∂vθ

∂t+

vr∂vθ

∂r

+vθ r

∂vθ

∂θ

+vφ

rsinθ

∂vθ

∂φ

+vrvθ

r−

v2 φcotθ

r

)

=−1 r

∂P ∂θ+

( 1 r3∂(r

3τ̃ r

θ)

∂r

+1

rsinθ

∂(τ̃

θθsinθ)

∂θ

+1

rsinθ

∂τ̃ φ

θ

∂φ

+τ̃ θ

r−

τ̃ rθ

r−

τ̃ φφcotθ

r

) +ρgθ

ρ

( ∂vφ

∂t

+vr∂vφ

∂r

+vθ r

∂vφ

∂θ

+vφ

rsinθ

∂vφ

∂φ

+vrvφ

r+

vφvθcotθ

r

)

=−

1

rsinθ

∂P

∂φ

+

( 1 r3∂(r

3τ̃ r

φ)

∂r

+1

rsinθ

∂(τ̃

θφsinθ)

∂θ

+1

rsinθ

∂τ̃ φ

φ

∂φ

+τ̃ φ

r−

τ̃ rφ

r+

τ̃ φθcotθ

r

) +ρgφ

EquationofM

otionforincompressible,New

tonianfluid

(Navier-Stokesequation)3compon

ents

inCartesian

coordinates

ρ

( ∂vx

∂t

+vx∂vx

∂x

+vy∂vx

∂y

+vz∂vx

∂z

)=

−∂P

∂x

+μ

( ∂2vx

∂x2+

∂2vx

∂y2+

∂2vx

∂z2

) +ρgx

ρ

( ∂vy

∂t+

vx∂vy

∂x

+vy∂vy

∂y

+vz∂vy

∂z

)=

−∂P ∂y+

μ

( ∂2vy

∂x2+

∂2vy

∂y2+

∂2vy

∂z2

) +ρgy

ρ

( ∂vz

∂t+

vx∂vz

∂x

+vy∂vz

∂y

+vz∂vz

∂z

)=

−∂P ∂z+

μ

( ∂2vz

∂x2+

∂2vz

∂y2+

∂2vz

∂z2

) +ρgz

Equation

ofM

otion

forincompressible,New

tonianfluid

(Navier-Stokesequation),

3compon

ents

incylin-

dricalcoordinates

ρ

( ∂vr

∂t+

vr∂vr

∂r

+vθ r

∂vr

∂θ

−v2 θ r+

vz∂vr

∂z

)=

−∂P ∂r+

μ

( ∂ ∂r

( 1 r∂(rvr)

∂r

) +1 r2∂2vr

∂θ2−

2 r2∂vθ

∂θ

+∂2vr

∂z2

) +ρgr

ρ

( ∂vθ

∂t+

vr∂vθ

∂r

+vθ r

∂vθ

∂θ

+vrvθ

r+

vz∂vθ

∂z

)=

−1 r

∂P ∂θ+

μ

( ∂ ∂r

( 1 r∂(rvθ)

∂r

) +1 r2∂2vθ

∂θ2+

2 r2∂vr

∂θ

+∂2vθ

∂z2

) +ρgθ

ρ

( ∂vz

∂t+

vr∂vz

∂r

+vθ r

∂vz

∂θ

+vz∂vz

∂z

)=

−∂P ∂z+

μ

( 1 r∂ ∂r

( r∂vz

∂r

) +1 r2∂2vz

∂θ2+

∂2vz

∂z2

) +ρgz

EquationofM

otionforincompressible,New

tonianfluid

(Navier-Stokesequation),3compon

ents

inspherical

coordinates ρ

( ∂vr

∂t+

vr∂vr

∂r

+vθ r

∂vr

∂θ

+vφ

rsinθ

∂vr

∂φ

−v2 θ+

v2 φ

r

)

=−∂P ∂r+

μ

( ∂ ∂r

( 1 r2∂ ∂r

( r2vr

))+

1

r2sinθ

∂ ∂θ

( sinθ∂vr

∂θ

) +1

r2sin2θ

∂2vr

∂φ2

−2

r2sinθ

∂ ∂θ(v

θsinθ)−

2

r2sinθ

∂vφ

∂φ

) +ρgr

ρ

( ∂vθ

∂t+

vr∂vθ

∂r

+vθ r

∂vθ

∂θ

+vφ

rsinθ

∂vθ

∂φ

+vrvθ

r−

v2 φcotθ

r

)

=−1 r

∂P ∂θ+μ

( 1 r2∂ ∂r

( r2∂vθ

∂r

) +1 r2

∂ ∂θ

(1

sinθ

∂ ∂θ(v

θsinθ)) +

1

r2sin2θ

∂2vθ

∂φ2

+2 r2∂vr

∂θ

−2cotθ

r2sinθ

∂vφ

∂φ

) +ρgθ

ρ

( ∂vφ

∂t

+vr∂vφ

∂r

+vθ r

∂vφ

∂θ

+vφ

rsinθ

∂vφ

∂φ

+vrvφ

r+

vφvθcotθ

r

)

=−

1

rsinθ

∂P

∂φ

+μ

( 1 r2∂ ∂r

( r2∂vφ

∂r

) +1 r2

∂ ∂θ

(1

sinθ

∂ ∂θ(v

φsinθ)) +

1

r2sin2θ

∂2vφ

∂φ2

+2

r2sinθ

∂vr

∂φ

+2cotθ

r2sinθ

∂vθ

∂φ

) +ρgφ

Note:

ther-com

pon

entof

theNavier-Stokesequationin

spherical

coordinates

may

besimplified

byad

ding0=

2 r∇

·vto

thecompon

entshow

nab

ove.

Thisterm

iszero

dueto

thecontinuityequation(m

assconservation).

See

Birdet.al.

Referen

ces:

1.R.B.Bird,W

.E.Stewart,

andE.N.Lightfoot,Transport

Phenomena,2n

ded

ition,W

iley:NY,2002.

2.R.B.Bird,R.C.Arm

strong,

andO.Hassager,

DynamicsofPolymericFluids:

Volume1Fluid

Mechanics,

Wiley:NY,1987.

    16 April 2014 

The Newtonian Constitutive Equation in Cartesian, Cylindrical, and Spherical coordinates 

Prof. Faith A. Morrison, Michigan Technological University 

     

Cartesian Coordinates 

̃ ̃ ̃̃ ̃ ̃̃ ̃

2

2

2

 

Cylindrical Coordinates 

̃ ̃ ̃̃ ̃ ̃̃ ̃

2 11 2 1 1

1 2

 

Spherical Coordinates 

̃ ̃ ̃̃ ̃ ̃̃ ̃

2 1 1sin1 2 1 sin sin

1sin

1sin

sinsin

1sin 2

1sin

cot

 

     

These expressions are general and are applicable to three‐dimensional flows.  For unidirectional flows they reduce to Newton’s law of Viscosity.  Reference:  Faith A. Morrison, An Introduction to Fluid Mechanics (Cambridge University Press: New York, 2013) 

hydraulic diam

eter general

, 4

PoRe

1 4.0log

Re Po duct 16

0.40Ge

ometry

PoCircle

16Equilateraltria

ngle

13.33

Slit

24Ellipse

(a,b)

32___________________________________________________________________________

void fraction,

empty bed

volume

total bed volum

etotal pa

rtical surface

areaparticle

volume

superficial vel

ocity,/

hydraulic diam

eter for packed

bed, 4 1

Reynolds num

ber for packed

bed /

friction factor

for packed be

d2

Mecha

nism

,

,

Cond

ensin

gsteam

1000

5000

5700

28,000

Cond

ensin

gorganics

200500

1100

2800

Boiling

liquids

30050

001700

28,000

Movingwater

503000

28017

,000

Movinghydrocarbo

ns10

300

551700

Stillair

0.54

2.823

Movingair

210

11.3

55Re

ference:

C.J.Ge

ankoplis,

Mag

nitude

ofSomeHe

atTran

sfer

Coefficients,pa

ge24

1

TableA.316

Geankoplis,

2003

1

TheEq

uatio

nof

Energy

inCartesian,cylindrical,and

sphe

ricalcoordinatesfor

New

tonian

fluidso

fcon

stantd

ensity,with

source

term

.Source

couldbe

electricalen

ergy

dueto

curren

tflow,che

micalen

ergy,etc.Tw

ocasesa

represen

ted:

thegene

ralcasewhe

rethermal

cond

uctiv

itymay

beafunctio

nof

tempe

rature

(vectorflux

appe

ars intheeq

uatio

ns);andthe

moreusualcase,whe

rethermalcond

uctiv

ityisconstant.

Fall20

13Faith

A.Morrison

,MichiganTechno

logicalU

niversity

Microscop

icen

ergy

balance,interm

sofflux;Gibb

snotation

Microscop

icen

ergy

balance,interm

sofflux;Ca

rtesiancoordinates

Microscop

icen

ergy

balance,interm

sofflux;cylindricalcoordinates

Microscop

icen

ergy

balance,interm

sofflux;sphe

ricalcoordinates

Fourier’s

lawof

heat

cond

uctio

n,Gibb

snotation:

Fourier’s

lawof

heat

cond

uctio

n,Ca

rtesiancoordinates:

Fourier’s

lawof

heat

cond

uctio

n,cylindricalcoordinates:

Fourier’s

lawof

heat

cond

uctio

n,sphe

ricalcoordinates:

2

TheEq

uatio

nof

Energy

forsystemsw

ithconstant

Microscop

icen

ergy

balance,constant

thermalcond

uctiv

ity;G

ibbs

notatio

n

Microscop

icen

ergy

balance,constant

thermalcond

uctiv

ity;C

artesia

ncoordinates

Microscop

icen

ergy

balance,constant

thermalcond

uctiv

ity;cylindricalcoordinates

Microscop

icen

ergy

balance,constant

thermalcond

uctiv

ity;sph

ericalcoordinates

Reference:

F.A.

Morrison

,“Web

Appe

ndixto

AnIntrod

uctio

nto

FluidMecha

nics,”Cambridge

University

Press,New

York,201

3.Ontheweb

atwww.che

m.m

tu.edu

/~fm

orriso/IFM_W

ebAp

pend

ixCD

2013

.pdf

A.2-11

('C)

0

15.6

26.7

37.8

65.6

93.3

121.l

148.9

204.4

260.0

315.6

A.2-11

T ("F)

32

60

80

100

150

200

250

300

400

500

600

Heat-Transfer Properties of Liquid Water, SI Units

µ. X JO3 (gf3p'Iµ.') T p er (Pa•s, or k /3 X JO' X J0-

8

(K) (kglm3) (kl/kg· K) kglm · s) (Wlm •K) N,, (1/K) (I/[(. m')

273.2 999.6 4.229 1.786 0.5694 13.3 -0.630

288.8 998.0 4.187 1.131 0.5884 8.07 1.44 10.93

299.9 996.4 4.183 0.860 0.6109 5.89 2.34 30.70

311.0 994.7 4.183 0.682 0.6283 4.51 3.24 68.0

338.8 981.9 4.187 0.432 0.6629 2.72 5.04 256.2

366.5 962.7 4.229 0.3066 0.6802 1.91 6.66 642

394.3 943.5 4.271 0.2381 0.6836 1.49 8.46 1300

422.l 917.9 4.312 0.1935 0.6836 1.22 10.08 2231

477.6 858.6 4.522 0.1384 0.6611 0.950 14.04 5308

533.2 784.9 4.982 0.1042 0.6040 0.859 19.8 11030

588.8 679.2 6.322 0.0862 0.5071 1.07 31.5 19 260

Heat-Transfer Properties of Liquid Water, English Units

p Cp µ. X JO3 k (gf3p'Iµ.')

(lb"') ( btu ) (�) ( btu ) /3 X JO' X 10-6

ft' lb111 • °F ft·S h•ft•'F N,, (ll'R) (1/'R. Ji')

62.4 1.01 1.20 0.329 13.3 -0.350

62.3 1.00 0.760 0.340 8.07 0.800 17.2

62.2 0.999 0.578 0.353 5.89 1.30 48.3

62.1 0.999 0.458 0.363 4.51 1.80 107

61.3 1.00 0.290 0.383 2.72 2.80 403

60.1 1.01 0.206 0.393 1.91 3.70 1010

58.9 1.02 0.160 0.395 1.49 4.70 2045

57.3 1.03 0.130 0.395 1.22 5.60 3510

53.6 1.08 0.0930 0.382 0.950 7.80 8350

49.0 1.19 0.0700 0.349 0.859 11.0 17 350

42.4 1.51 0.0579 0.293 1.07 17.5 30 300

A.3-3 Physical Properties ol' Air at 101.325 kPa (1 Atm Abs), SI Units

µ. X JO5 T p c,, (Pa·s.or k /3 X JO' gf3p"Iµ."

("C) (/() (kglnr1 ) (/cJ!kg · K) kg/111 · s) (W/m·K) Nr, (//K) (//K · 1113)

-17.8 255.4 1.379 1.0048 1.62 0.02250 0.720 3.92 2.79 X 108

0 273.2 1.293 1.0048 1.72 0.02423 0.715 3.65 2.04 X 108

10.0 283.2 1.246 1.0048 1.78 0.02492 0.713 3.53 1.72 X 108

37.8 311.0 1.137 1.0048 1.90 0.02700 0.705 3.22 1.12 X 108

65.6 338.8 1.043 1.0090 2.03 0.02925 0.702 2.95 0.775 X 108

93.3 366.5 0.964 1.0090 2.15 0.03115 0.694 2.74 0.534 X 108

121.1 394.3 0.895 1.0132 2.27 0.03323 0.692 2.54 0.386 X 108

148.9 422.l 0.838 1.0174 2.37 0.03531 0.689 2.38 0.289 X 108

176.7 449.9 0.785 1.0216 2.50 0.0372 l 0.687 2.21 0.214 X 108

204.4 477.6 0.740 1.0258 2.60 0.03894 0.686 2.09 0.168 X 108

232.2 505.4 0.700 l.0300 2.71 0.04084 0.684 1.98 0.130 X 108

260.0 533.2 0.662 1.0341 2.80 0.04258 0.680 1.87 0.104 X 108

A.3-3 Physical Properties of Air at 101.325 kPa (1 Atm Abs), English Units

p c,, k T

Cb

''.') ( bw ) µ. (

btu ) /3 X JO' g{3p'Iµ.

'("F) ft·' lb111 • °F ( centipoise) h·ft·"F N,, (ll"R) (I!"R ·fr')

0 0.0861 0.240 0.0162 0.0130 0.720 2.18 4.39 X 106

32 0.0807 0.240 0.0172 0.0140 0.715 2.03 3.21 X 106

50 0.0778 0.240 0.0178_ 0.0144 0.713 1.96 2.70 X 106

100 0.0710 0.240 0.0190 0.0156 0.705 1.79 1.76 X 106

150 0.0651 0.241 0.0203 0.0169 0.702 1.64 1.22 X 106

200 0.0602 0.241 0.0215 0.0180 0.694 1.52 0.840 X 106

250 0.0559 0.242 0.0227 0.0192 0.692 1.41 0.607 X 106

300 0.0523 0.243 0.0237 0.0204 0.689 1.32 0.454 X 106

350 0.0490 0.244 0.0250 0.0215 0.687 1.23 0.336 X 106

400 0.0462 0.245 0.0260 0.0225 0.686 1.16 0.264 X 106

450 0.0437 0.246 0.0271 0.0236 0.674 1.10 0.204 X 106

500 0.0413 0.247 0.0280 0.0246 0.680 1.04 0.163 X 106

Source: National Bureau of Stand,1rds, Circufnr 461C, 1947; 564, 1955: NBS-NACA. Ta hies of Thermnl Properries of Gnses, 1949; F. G. Keyes. Tmns. A.S.I\I. £., 73,590.597 { !95 l ): 74, !J03 { !952); D. D. Wngmnn, Selected Values of Chemical Thermody11amic Properties. Washington, D.C.: National Bureau of Stnndards, ! 953.

1

Heat

Tran

sfer

Data

Correlations

forE

xaminations

CM31

10TransportP

heno

men

aI

Mich

igan

Techno

logicalU

niversity

ProfessorF

aith

A.Morrison

1De

cembe

r202

0

I. Forced

Conv

ectio

nTh

roug

hPipe

s

Inforced

convectio

n,wede

term

ined

from

dimen

sionalanalysis

that

theNu

sseltn

umbe

risa

functio

nof

atmostR

e,Pr,

/,andviscosity

ratio

.

Prandtlnum

ber

(fluidprop

ertie

s)Pr

(1)

Inpipe

flowwith

heat

transfer

taking

place,theflu

iden

tersat

bulkflu

idtempe

rature

andexits

at.

isthetempe

rature

ofthewall.ForN

udata

correlations

inforced

convectio

nthrough

pipe

s,allfluidmaterialprope

rtiese

xcep

tareevaluatedat

themeanbu

lktempe

rature.

Themeanbu

lktempe

rature

isgivenby

Meanbu

lktempe

rature

2(2)

A. Laminar

Flow

inPipe

s

Sied

erandTate’scorrelation(Geankop

lis,p26

0)forlam

inar

flowis

Laminar

flow

Nu1.86R

ePr.

(3)

(4)

Arith

meticmean

drivingforce

2(5)

B. Tu

rbulen

tFlowinPipe

s

Sied

erandTate’scorrelation(Geankop

lis,p26

1)forturbu

lent

flowis

Turbulen

tflow

Nu0.027R

e. Pr.

(6)

(7)

2

Logmeandriving

force

(8)

II.

Forced

Convectio

nArou

ndtheOu

tsideof

aCylin

der

Inhe

attransfer

taking

placebe

tweenaflu

idat

bulktempe

rature

flowingpe

rpen

diculartoa

cylinde

rwith

walltem

perature

,the

materialprope

rtiesa

reevaluatedat

thefilm

tempe

rature.

Film

tempe

rature

2(9)

Thedata

correlationforN

usseltnu

mbe

rinthiscase

is

Outsid

eCylinde

rNu

RePr(10)

Wallb

ulkdriving

force

(11)

Thevalues

ofand

depe

ndon

theRe

ynolds

numbe

r(Ge

ankoplis,

Table4.61,p2

72).These

values

arevalid

forPr

0.6.Re

mC

140.330

0.989440

0.3850.911

404,000

0.4660.683

4,000410

0.6180.193

4102.51

00.805

0.0266

3

III.

NaturalCon

vectionfro

mVa

rious

Geom

etrie

s

Naturalcon

vectionhe

attransfer

coefficientsfrom

vario

ussurfa

cesh

avebe

enfoun

dby

dimen

sional

analysisandexpe

rimen

tally

tocorrelateas

follows:

Naturalcon

vection

(various

geom

etrie

s)Nu

Gr Pr(12)

Grasho

fnum

ber

Gr(13)

Thevalues

for

and

depe

ndon

thegeom

etry;value

smay

befoun

dinGe

ankoplisinTable4.71

(p27

8,show

nbe

low).Table4.72(p28

0,ne

xtpages)provides

simplified

versions

ofthecorrelations

specialized

tocommon

fluids(air,water,organicliquids).

4

Reference:

C.J.Ge

ankoplis,

TransportP

rocesses

andSeparatio

nProcessP

rincip

les,4t

hEd

ition

,Pren

ticeHa

ll,20

03.

Table 1:  Emissivity 𝜺 of solids (300K)   Material  𝜺 

Aluminum foil  0.04 Asbestos board  0.96 Polished brass  0.03 

Cast iron, turned and heated  0.60‐0.70 Concrete  0.85 

Ice, smooth  0.966 Ice, rough  0.985 Plaster  0.98 

Roofing paper  0.91 Sand  0.76 

Steel, Oxidized  0.79 Wrought Iron  0.94 

 

Reference:  Engineering Toolbox, www.engineeringtoolbox.com/emissivity‐coefficients‐d_447.html  

 

 

 

 

 

 

 

 

 

 

 

Reference:  Geankoplis, 4th Edition 

 

𝜎 0.1712 10 𝐵𝑇𝑈ℎ 𝑓𝑡 𝑅

𝜎 5.676 10 𝑊𝑚 𝐾

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Emissivity ε of solids and other properties v4b