Mec 551 Convection

7/24/2019 Mec 551 Convection

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EC 551

EC 551

THER AL ENGINEERING

HER AL ENGINEERING

EC 551

EC 551

THER AL ENGINEERINGHER AL ENGINEERING

3.0 Convection

.0 Convection

.0 Convection

.0 Convection


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Convection AnalysisConvection Analysis

Convection is similar to conduction in that it requiresConvection is similar to conduction in that it requires

the presence of a material medium but differentthe presence of a material medium but different

because it also requires the presence of fluid motion.because it also requires the presence of fluid motion.


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Convection AnalysisConvection Analysis

Fluid motion enhances heatFluid motion enhances heat

transfer, because it initiatestransfer, because it initiates

higher rates of conduction byhigher rates of conduction by

bringing more hot and coldbringing more hot and coldmolecules into contactmolecules into contact

Heat transfer through a liquid orHeat transfer through a liquid or

gas can be either by conductiongas can be either by conductionor convection. Conduction is theor convection. Conduction is the

limiting case of no fluid motion.limiting case of no fluid motion.

Convection involves bothConvection involves both

conduction and fluid motionconduction and fluid motion..


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3.13.1 Convective !rinciplesConvective !rinciples


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Convection !rinciplesConvection !rinciples

#here are#here are t$ot$otypes of convection%types of convection%

&atural or Free Convection%&atural or Free Convection%

Fluid motion is caused by naturalFluid motion is caused by natural

means such as the buoyancymeans such as the buoyancyeffect, $hich manifests itself aseffect, $hich manifests itself as

the rise of $armer air and the fallthe rise of $armer air and the fall

of cooler air.of cooler air.

Forced Convection%Forced Convection%

Fluid is forced to flo$ over aFluid is forced to flo$ over asurface by e'ternal means (suchsurface by e'ternal means (such

as a pump or fan).as a pump or fan).


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#here are#here are t$ot$otypes oftypes of

forced convectionforced convection%%

+'ternal%+'ternal% Fluid is forced to flo$Fluid is forced to flo$over a surface.over a surface.

nternal%nternal%

Fluid is forced to flo$ inFluid is forced to flo$ in

a pipe or channel.a pipe or channel.

+-#+.

&A/

+-#+.

&A/

,+.&A/

,+.&A/


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#he difference bet$een e'ternal and internal flo$s is

sho$n in the figure belo$%

+'ternal Flo$+'ternal Flo$

nternalnternal

Flo$Flo$


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Convection is described by &e$tons /a$ of Cooling%Convection is described by &e$tons /a$ of Cooling%

Convection heat transfer coefficient (h)Convection heat transfer coefficient (h)

efined as the rate of heat transfer bet$een a solid surfaceefined as the rate of heat transfer bet$een a solid surface

and a fluid per unit surface area per unit temperatureand a fluid per unit surface area per unit temperaturedifference.difference.

Convection !rinciplesConvection !rinciples(&e$tons /a$ of Cooling)(&e$tons /a$ of Cooling)

( )= TTAhQ ssconv


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Convection !rinciplesConvection !rinciples(&usselt &umber)(&usselt &umber)

&usselt &umber&usselt &umber

eveloped by 5ilhelm &usselteveloped by 5ilhelm &usselt

(12614"0) from 7ermany(12614"0) from 7ermany

n convection analysis, it isn convection analysis, it iscommon practice to non6common practice to non6

dimensionali8ed the governingdimensionali8ed the governing

equations and combine theequations and combine the

variables, $hich group together invariables, $hich group together indimensionless numbers 9 to reducedimensionless numbers 9 to reduce

the number of variables.the number of variables.


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#he &usselt number is a#he &usselt number is a non6dimensionali8ed hnon6dimensionali8ed h,,

defined as%defined as%

k

hLNu c= //cc 6 Characteristic /ength6 Characteristic /ength

; 6 #hermal conductivity of fluid; 6 #hermal conductivity of fluid


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#a;ing the ratio of these t$o equations%#a;ing the ratio of these t$o equations%

#hus &u represents the enhancement of heat transfer through a#hus &u represents the enhancement of heat transfer through a

fluid layer as a result of convection relative to conductionfluid layer as a result of convection relative to conduction

across the same fluid layer. #he larger &u, the more effectiveacross the same fluid layer. #he larger &u, the more effective

the convection.the convection.

&u> 1 for a fluid layer, represents pure conduction.&u> 1 for a fluid layer, represents pure conduction.

Nu

k

LhTh

q

q

LTkcond

conv =

=

=


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Convection !rinciplesConvection !rinciples(?iscosity)(?iscosity)

?iscosity?iscosity

A measure of the internal stic;iness ofA measure of the internal stic;iness of

the fluid. #he friction force bet$een t$othe fluid. #he friction force bet$een t$o

fluid layers moving relative to onefluid layers moving relative to one

another. Caused by the cohesive forcesanother. Caused by the cohesive forcesbet$een the molecules in the liquidsbet$een the molecules in the liquids

and by the molecular collisions in theand by the molecular collisions in the

gases.gases.

#here are t$o e'pressions for viscosity%#here are t$o e'pressions for viscosity%

ynamic viscosity (or absoluteynamic viscosity (or absolute

viscosity),viscosity),

@inematic viscosity,@inematic viscosity,


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ynamic viscosityynamic viscosity((

) 9) 9 #he shear force per unit area#he shear force per unit arearequired to drag on layer of fluid $ith unit velocity passedrequired to drag on layer of fluid $ith unit velocity passed

another layer a unit distance a$ay from the fluid.another layer a unit distance a$ay from the fluid.

@inematic viscosity@inematic viscosity(() 9) 9 #he ratio of dynamic viscosity#he ratio of dynamic viscosityto density.to density.


=

dydu

=


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?iscous flo$s?iscous flo$s

Flo$s in $hich the effects ofFlo$s in $hich the effects of

viscosity are significant.viscosity are significant.

nviscid flo$snviscid flo$s

Flo$s in $hich the effects ofFlo$s in $hich the effects of

viscosity is small and can beviscosity is small and can be

neglected $ithout much loss inneglected $ithout much loss in

accuracy. Frictionless oraccuracy. Frictionless or

ideali8ed flo$s.ideali8ed flo$s.


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Convection !rinciplesConvection !rinciples(Compressibility)(Compressibility)

Compressible flo$Compressible flo$

7ases are highly compressible, meaning7ases are highly compressible, meaning

that there is a significant density change ofthat there is a significant density change of

fluid during flo$ (e.g. air).fluid during flo$ (e.g. air).

ncompressible flo$ncompressible flo$

ensities that are essentially constant,ensities that are essentially constant,

such as many liquids (e.g. $ater).such as many liquids (e.g. $ater).

7as7as

/iquid/iquid


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Convection !rinciplesConvection !rinciples(#ypes of Flo$s)(#ypes of Flo$s)

/aminar Flo$/aminar Flo$

Highly ordered fluid motionHighly ordered fluid motion

such as the flo$ of highlysuch as the flo$ of highly

viscosity fluids li;e oil at lo$viscosity fluids li;e oil at lo$

velocities.velocities.

#urbulent Flo$#urbulent Flo$

Highly disordered (orHighly disordered (or

chaotic) flo$ that typicallychaotic) flo$ that typically

occurs at high velocities.occurs at high velocities.


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niform Flo$niform Flo$

&o change in the fluid velocity or volume over a specified&o change in the fluid velocity or volume over a specified

region.region.


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3.23.2 Convection boundaryConvection boundarylayer theorylayer theory


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Convection !rinciplesConvection !rinciples(?elocity oundary /ayer)(?elocity oundary /ayer)

A velocity boundary layer can beA velocity boundary layer can bedefineddefined

&o slip condition&o slip condition 5hen the fluid is forced to flo$ over a5hen the fluid is forced to flo$ over a

solid surface that is non6porous (e.g.solid surface that is non6porous (e.g.impermeable fluid), it is observed that theimpermeable fluid), it is observed that thefluid in motion comes to a complete stopfluid in motion comes to a complete stopat the surface and there is no slip.at the surface and there is no slip.

ecause the fluid layer adDacent to the $allecause the fluid layer adDacent to the $allstic;s (due to friction), it slo$s the ne'tstic;s (due to friction), it slo$s the ne'tlayer and so on.


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?elocity boundary development on a flat plate%?elocity boundary development on a flat plate%

#he boundary layer thic;ness (#he boundary layer thic;ness (dd) is normally defined) is normally defined

as $here%as $here%

= uu 99.0


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#he dashed line, divides the flo$#he dashed line, divides the flo$

over the plate into t$o regions%over the plate into t$o regions%

oundary layer regionoundary layer region

n $hich the viscous effects andn $hich the viscous effects and

velocity changes are significant.velocity changes are significant.

nviscid flo$ regionnviscid flo$ region n $hich the friction effects aren $hich the friction effects are

negligible and the velocitynegligible and the velocity

remains constant.remains constant.

uEyy

''

Heated


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Flo$ regions in velocity boundary of a flat plate%Flo$ regions in velocity boundary of a flat plate%


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Comparison of a laminar and turbulent velocityComparison of a laminar and turbulent velocity

boundary layer profile%boundary layer profile%


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Convection !rinciplesConvection !rinciples(#hermal oundary /ayer)(#hermal oundary /ayer)

/i;e$ise there is a/i;e$ise there is a thermalthermal

boundary layerboundary layer

&o temperature Dump condition&o temperature Dump condition

ecause velocity of the fluidecause velocity of the fluid

is 8ero at the point of contactis 8ero at the point of contact

$ith the solid surface, the$ith the solid surface, the

fluid and solid surface mustfluid and solid surface musthave the same temperaturehave the same temperature

at the point of contact.at the point of contact.

yy

''

#E

Heated


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Convection !rinciplesConvection !rinciples(#hermal oundary /ayer)(#hermal oundary /ayer)

#hermal boundary development on a flat plate%#hermal boundary development on a flat plate%

#he thic;ness of the thermal boundary layer (#he thic;ness of the thermal boundary layer (ddtt) at any location) at any location

along the surface is defined as the distance from the surface atalong the surface is defined as the distance from the surface at

$hich%$hich%

DD#>#6##>#6#ss>:.44>:.44(#(#6#6#ss))

#sB:.44(#6#s)


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Convection !rinciplesConvection !rinciples(!randtl &umber)(!randtl &umber)

!randtl &umber!randtl &umber

eveloped by /ud$ig !randtl (10"614"3) ofeveloped by /ud$ig !randtl (10"614"3) of

7ermany.7ermany.

#he relative thic;ness of the velocity and#he relative thic;ness of the velocity and

thermal boundary layers is best described bythermal boundary layers is best described by

a dimensionless !randtl number (belo$)%a dimensionless !randtl number (belo$)%

k

C

HeatofyDiffusivitMolecular

MomentumofyDiffusivitMolecular

p==

=

Pr


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Convection !rinciplesConvection !rinciples(eynolds &umber)(eynolds &umber)

eynolds &umbereynolds &umber

erived by Gsbourne eynolds (1261412)erived by Gsbourne eynolds (1261412)

of ritainof ritain

#he transition from laminar to turbulent flo$#he transition from laminar to turbulent flo$depends on the surface geometry, surfacedepends on the surface geometry, surface

roughness, free stream velocity, surfaceroughness, free stream velocity, surface

temperature, and type of fluid (among othertemperature, and type of fluid (among other

things).things).

Ho$ever, the flo$ regime primarily dependsHo$ever, the flo$ regime primarily depends

upon the ratio of inertia forces to viscousupon the ratio of inertia forces to viscous

forces in a fluid. #his is a dimensionlessforces in a fluid. #his is a dimensionless

quantity, ;no$n asquantity, ;no$n as eynolds numbereynolds number(e).(e).


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#he eynolds number is defined as%#he eynolds number is defined as%

LVLV

ForcesViscousForcesnertia ===Re

? upstream velocity/ characteristic length

n > mrkinematic viscosity of fluid


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AA large elarge e(inertia forces large)(inertia forces large) Ieans that the viscous forces cannot contain random andIeans that the viscous forces cannot contain random and

rapid fluctuations (turbulent).rapid fluctuations (turbulent).

AA small esmall e(viscous forces large)(viscous forces large)

@eeps the fluid in6line (laminar).@eeps the fluid in6line (laminar).

#he eynolds number $here the flo$ becomes turbulent is#he eynolds number $here the flo$ becomes turbulent is

called the critical eynolds number (ecalled the critical eynolds number (ecritcrit))

LVLV

ForcesViscous

Forcesnertia =

==Re


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For flo$ over a flat plate, the generally acceptedFor flo$ over a flat plate, the generally accepted

value of evalue of ecritcritis%is%

Flat !late%Flat !late%

$here%$here% ''critcrit>> istance bet$een the leading edgeistance bet$een the leading edge

of the plate to the transition pointof the plate to the transition pointfrom laminar to turbulent flo$ ta;es place.from laminar to turbulent flo$ ta;es place.

5

105Re =

=

critcrit

!u


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3.33.3 Forced convection over anForced convection over an

e'terior surfacee'terior surface(laminar and turbulent flo$)(laminar and turbulent flo$)


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+'ternal Flo$+'ternal Flo$

#he convection equations for an e'ternal flo$ can be#he convection equations for an e'ternal flo$ can be

derived from thederived from the conservation of massconservation of mass,, conservationconservation

of energyof energy, and the, and the conservation of momentumconservation of momentum

equations.equations.


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+'ternal Flo$ +quations+'ternal Flo$ +quations(Conservation of Iass)(Conservation of Iass)

Conservation of IassConservation of Iass

d!!

u

u

+

d'd'

dydy

dydy

dvv

+

u

v

( )

( )Area"nit

y

Area"nit

!

d!vm

dyum

1

1

=

=


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+'ternal Flo$ +quations+'ternal Flo$ +quations(Conservation of Iass)(Conservation of Iass)

ate of massate of mass

flo$ intoflo$ into

control volumecontrol volume


flo$ intoflo$ into



flo$ out offlo$ out of



flo$ out offlo$ out of


>>

dyd!yvd!vdyd!

!udyud!vdyu

d!dyy

vvdyd!

!

uud!vdyu

++

+=+

++

+=+

0=

+

y

v

!

uJ 26 Continuity +quationJ 26 Continuity +quation


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+'ternal Flo$ +quations+'ternal Flo$ +quations(Conservation of Iomentum)(Conservation of Iomentum)

Conservation of IomentumConservation of Iomentum

KKmma > &et Forcea > &et Force

d!!

##

+#

dyy

+

d'd'

dydy


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+'ternal Flo$ +quations+'ternal Flo$ +quations(Conservation of Iomentum)(Conservation of Iomentum)

n the '6direction%n the '6direction%

n the y6direction%n the y6direction%

volumeunitper force$ody

!

forcesshearandviscousofeffectNet

forcepressureNet

du

%y

u

!

u

!

#

y

uv

!

uu +

+

+

=

+

2

2

2

2

volumeunitperforce$ody

y

forcesshearandviscousofeffectNet

force

pressureNetdv

%y

v

!

v

y

#

y

vv

!

vu +

+

+

=

+

2

2

2

2


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+'ternal Flo$ +quations+'ternal Flo$ +quations(Conservation of +nergy)(Conservation of +nergy)

Conservation of +nergyConservation of +nergy

d'd'

dydy

++heat out, yheat out, y ++mass out, ymass out, y

++mass in, ymass in, y++heat in, yheat in, y

++mass in, 'mass in, '

++heat in, 'heat in, '

++mass out, 'mass out, '

++heat out, 'heat out, '

0= outin &&


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+'ternal Flo$ +quations+'ternal Flo$ +quations(Conservation of +nergy)(Conservation of +nergy)

7eneral 26 energy equation7eneral 26 energy equation

For 26 inviscid flo$%For 26 inviscid flo$%

222

2

2

2

2

2

+

+

+

+

+

=

+

!

v

y

u

y

v

!

u

y

T

!

Tk

y

Tv

!

TuCp

+

=

+

2

2

2

2

yT

!Tk

yTv

!TuCp


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Convection over a Flat !lateConvection over a Flat !late

Consider laminar flo$ over a flat plat. 5hen viscousConsider laminar flo$ over a flat plat. 5hen viscousdissipation is negligible, the convection equationsdissipation is negligible, the convection equations

reduce for steady, incompressible laminar flo$ ($ithreduce for steady, incompressible laminar flo$ ($ith

constant properties) over a flat plate.constant properties) over a flat plate.

'

y

##

, u, u

u(',:)> :u(',:)> :

v(',:)> :v(',:)> :

#(',:)> ##(',:)> #ss

dydy

d'd'

oundary layeroundary layer


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Consider elemental control volume for force balanceConsider elemental control volume for force balance

in the laminar boundary layer.in the laminar boundary layer.

Continuity%Continuity%

Iomentum%Iomentum%

+nergy%+nergy%

0

=

+

y

v

!

u

2

2

y

u

y

uv

!

uu

=

+

2

2

y

T

y

Tv

!

Tu

=

+


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ecall, that the stream function is defined as%ecall, that the stream function is defined as%

ependent variable%ependent variable%

!v

yu

=

=

;

( )yu

u!u

f

=

=


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#herefore%#herefore%

=

=

=

=

=

=

=

=

fd

df

!

u

f!u

u

d!

df

u

!u

!!v

d

dfu

!

u

d

df

u

!u

yyu

2

1

2


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Convection over a Flat !lateConvection over a Flat !late(Iomentum +quation)(Iomentum +quation)


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sing the definitions for f andsing the definitions for f and M, the boundary equations inM, the boundary equations in

terms of the similarity variables can be found.terms of the similarity variables can be found.

( )

1

0

00

0

=

=

=

=

=

d

df

d

df

f Ho$ever, the transformed equationHo$ever, the transformed equation

$ith its similarity variable cannot be$ith its similarity variable cannot besolved analytically.solved analytically.

#herefore, an alternative solution is#herefore, an alternative solution is

necessary.necessary.

C ti Fl t !l tC ti Fl t !l t


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#he non6dimensional velocity profile can be obtained by#he non6dimensional velocity profile can be obtained by

plotting uuplotting uu

vs.vs. MM. #he results agree e'perimentally.. #he results agree e'perimentally.

A value of% corresponds to%A value of% corresponds to%

ecall that the definition of a velocity boundary layer is $hen%ecall that the definition of a velocity boundary layer is $hen%

992.0==u

u

d

df

0.5=

99.0=uu



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+nergy +quation+nergy +quation

@no$ing the velocity profile, $e can no$ solve the energy@no$ing the velocity profile, $e can no$ solve the energy

equation.equation.

ntroduce dimensionless temperature%ntroduce dimensionless temperature%

&ote% both #&ote% both #ssand #and #

are constant.are constant.

Convection over a Flat !lateConvection over a Flat !late(+nergy +quation)(+nergy +quation)

( ) ( )

s

s

TT

Ty!Ty!

=

,,



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N(M)


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and%and% ( )

yud

df

yu

u

!u

f

=

=

=

Con ection o er a Flat !lateConvection over a Flat !late


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""



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=

+

!

u

d

d

!

u

yu!

u

u!

u

!

u

!

u

ud

d

2

2

1

2

1

2

2

21

dd

u!

!y!u

!!u

udd =

+



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2

2

2

d

d

yu!

u

u!

u

u

!

d

d=

+

2

2

Pr

2

d

d

yud

d

f

=

0Pr22

2

=+

d

df

d

d

=Pr

!randtl number!randtl number

+L& *6"

te't



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A closed form solution cannot be obtained for this boundaryA closed form solution cannot be obtained for this boundary

layer problem, and it must be solved numerically.layer problem, and it must be solved numerically.

f this equation is solved for numerous values of !r, then forf this equation is solved for numerous values of !r, then for

!r O :.*, the non6dimensional temperature gradient at the!r O :.*, the non6dimensional temperature gradient at the

surface is found to be (reference #able *63, p. 3" in te't)%surface is found to be (reference #able *63, p. 3" in te't)%

31

Pr332.00 ==

d

d



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#he temperature gradient at the surface is%#he temperature gradient at the surface is%


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#herefore the#herefore the locallocalconvection coefficient and &usselt numberconvection coefficient and &usselt number

become%become%

( )[ ]

=

=

=

=

TT

TTk

TT

k

TT

qh

s

!

u

s

s

yyT

s

s

!

3

1

Pr332.00

!ukh! =

3

1

Pr332.0



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#he local &usselt number is the dimensionless temperature#he local &usselt number is the dimensionless temperature

gradient at the surface. #his is defined as%gradient at the surface. #his is defined as%

#hus for#hus for !r O :.*!r O :.*, the, the locallocal&usselt number&usselt number for laminar flo$for laminar flo$is%is%

Convection over a Flat !lateConvection over a Flat !late(/aminar Flo$)(/aminar Flo$)

k

!hNu !!

=

21

31

RePr332.0 =!Nu



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#he local friction coefficient (C#he local friction coefficient (CF'F') can also be determined.) can also be determined.


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#herefore the local s;in friction coefficient is%#herefore the local s;in friction coefficient is%

21

Re664.02

2,

=

= !

'all!F

u

C

2

,2

1= uC !F'all



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#he average heat transfer coefficient over the entire plate can be#he average heat transfer coefficient over the entire plate can be

obtained by integrating over its length%obtained by integrating over its length%

d!h

L

hL

!=

0

1

( )

L

k

LuL

k

!u

L

k

d!

!

u

L

kh

L

L

21

31

31

31

31

RePr664.0

Pr664.0

2Pr332.0

Pr332.0

0

0

=

=

=

=



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Convection over a Flat !lateConvection over a Flat !late(+'ample 3.1)(+'ample 3.1)

+'ample 3.1a+'ample 3.1a Calculate the velocity and theCalculate the velocity and the

thermal boundary layer thic;nessthermal boundary layer thic;ness P of the $ay alongP of the $ay along

a flat plate that is ": m long. 5ater (#a flat plate that is ": m long. 5ater (#sat Hsat H22GG> :> : QC)QC)

flo$s over it at ms. #he plate is ;ept at a surfaceflo$s over it at ms. #he plate is ;ept at a surface

temperature (#temperature (#ss> : QC).> : QC).

": m": m

''

y

: QC

ms

#s> :QC



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#he first step is to calculate the mean film temperature of the#he first step is to calculate the mean film temperature of the

fluid flo$ing along the plate.fluid flo$ing along the plate.

#his is Dust the average of the surface temperature and the fluid#his is Dust the average of the surface temperature and the fluidbul; temperature.bul; temperature.

CCCTT

T sfilm =+

=+

= 602

4080

2

": m": m''

yy

: QC

: QC



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*4


For liquid $ater at *:For liquid $ater at *: QC from #able A64 in the te't boo;%QC from #able A64 in the te't boo;%

": m": m''

yy

: QC

: QC

99.2Pr

654.0

67.4

3.983 3

=

=

=

=

Cm(

sm

k%

m

k%

k


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01


+'ample 3.1b+'ample 3.1b &o$ calculate the convective heat transfer.&o$ calculate the convective heat transfer.

First $e must chec; to see $hether the entire plate is in aFirst $e must chec; to see $hether the entire plate is in a

laminar boundary layer or not.laminar boundary layer or not.


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#herefore $e can use the follo$ing equation to find h%#herefore $e can use the follo$ing equation to find h%

( ) ( ) ( ) ( )

( ) ( )

Cm(

sm

k%

m

k%

sm

Cm(

m

!

uk

!

ukh

=

=

=

=

2

33

1

31

31

619.0

5067.4

3.9834654.099.2332.0

Pr332.0Pr332.0



73/174

03


sing this h, $e can no$ find the convection heat transfer%sing this h, $e can no$ find the convection heat transfer%

( ) ( )2

2

8.244080619.0

)(

m

(

Cm

(

s

CC

TThq

==

=



74/174

0

Convection over a Flat !lateConvection over a Flat !late(#urbulent and Ii'ed Flo$s)(#urbulent and Ii'ed Flo$s)

Turbulent

Completely #urbulent Flo$

Ii'ed /aminar#urbulent Flo$



75/174

0"

Convection over a Flat !lateConvection over a Flat !late(#urbulent and Ii'ed Flo$s)(#urbulent and Ii'ed Flo$s)

&ote&ote% if it had been found that the boundary layer $as not% if it had been found that the boundary layer $as notcompletely laminar another equation for h could have beencompletely laminar another equation for h could have been

used instead.used instead.

ForFor turbulent flo$turbulent flo$(all over the plate)%(all over the plate)%

For aFor a mi'ed combinationmi'ed combinationof laminar and turbulent flo$ over theof laminar and turbulent flo$ over theplate%plate%

75 10Re105

60Pr6.0

3

1

PrRe037.0 8.0 = LNu

( ) 31Pr871Re037.0 8.0 = LNu 75 10Re10560Pr6.0

L



76/174

0*

+'ample 3.2+'ample 3.2 Gil flo$s over a :6m long heated plate at freeGil flo$s over a :6m long heated plate at freestream conditions of " ms and 2"stream conditions of " ms and 2"QC. f the plate is held at "QC.QC. f the plate is held at "QC.

a) etermine the velocity and thermal boundary layera) etermine the velocity and thermal boundary layer

thic;nesses at the middle of the plate.thic;nesses at the middle of the plate.

b) Calculate the total heat flu' from the surface for a 16mb) Calculate the total heat flu' from the surface for a 16m

$idth.$idth.

c) Calculate the total convection heat transfer.c) Calculate the total convection heat transfer.


: m

uu

> " ms> " ms

##> 2"> 2"QCQC

##ss> "> "QCQC



77/174

00


First calculate the film temperature (#First calculate the film temperature (# ff))

From #ables for oil at 3"From #ables for oil at 3"QC, the fluid properties are%QC, the fluid properties are%

CCCTT

T sfilm =+

=+

= 352

4525

2

3

2

255,1

2864.0

105.3

711,3Pr

4

m

k%

Cm(

sm

k

=

==

=


78/174



79/174

04


#he hydrodynamic (or velocity) boundary layer is%#he hydrodynamic (or velocity) boundary layer is%

#he thermal boundary layer is%#he thermal boundary layer is%

( )cmorm

m!! 7.18187.0

1086.2

205

Re

5

520 =

=

==

( ) mmormm

t

8.110118.0711,3026.1

187.0

Pr026.1

31

31

==

=


80/174



81/174

1


c) sing thec) sing the mi'ed &u equationmi'ed &u equationfor a flat plate%for a flat plate%

( )

( )

[ ] ( )

7.600,9

711,38711071.5037.0

Pr871Re037.0

31

31

8.05

8.0

==

= LNu

( ) ( )Cm

(Cm(

m

L

kNuh

=

=

=

27.6840

2864.07.600,9



82/174

2


#he total heat flu' per is%#he total heat flu' per is%

( )

( ) ( ) ( )(

CCmm

TTAhQ

Cm

(

ss

960,54

25451407.68 2

=

=

=


83/174

Forced ConvectionForced Convection


84/174

Forced Convectiono ced Co ect o(on Cylinders and


85/174

"

Forced Convection(over Circular and &on6Circular Cylinders)(over Circular and &on6Circular Cylinders)

Additionally the follo$ing empirical correlations have been madeAdditionally the follo$ing empirical correlations have been madeby Su;aus;as and Ta;ob for the average &usselt number for flo$by Su;aus;as and Ta;ob for the average &usselt number for flo$

over circular and non6circular cylinders (#able 061 in te't)%over circular and non6circular cylinders (#able 061 in te't)%



86/174

*

Forced Convection(over Circular and &on6Circular Cylinders)(over Circular and &on6Circular Cylinders)



87/174

0

(+'ample 3.3)(+'ample 3.3)

+'ample 3.3+'ample 3.3 A long 1:6cm diameter he'agonal steam pipeA long 1:6cm diameter he'agonal steam pipe$hose e'ternal surface temperature is 11:$hose e'ternal surface temperature is 11:UC passes throughUC passes through

some open area that is not protectedsome open area that is not protected against the $ind.against the $ind.

etermine the rate of heat loss $hen the air is at 1 atmetermine the rate of heat loss $hen the air is at 1 atm

pressure and 1:pressure and 1:UC and the $ind is blo$ing across a 16m lengthUC and the $ind is blo$ing across a 16m length

of pipe at a velocity of ms.of pipe at a velocity of ms.

??> ms> ms

##

> 1:> 1:UCUC#s>11:UC

1: cm

1 m1 m



88/174

(+'ample 3.3)(+'ample 3.3)

#he properties of air at the average film temperature#he properties of air at the average film temperatureof%of%

can be found from #able A61" as%can be found from #able A61" as%

CCCTT

T sfilm =+

=+

= 602

10110

2

sm

Cm

(

k25

10896.1

7202.0Pr;02808.0

=

==



89/174

4

(+'ample 3.3)(+'ample 3.3)

#he eynolds number is%#he eynolds number is%

#he &usselt number can be determined from #able 061#he &usselt number can be determined from #able 061

in the te't boo;%in the te't boo;%

( ) ( ) 45

10219.410896.1

10.08Re

2 =

=

=

sm

sm mDV

( ) ( )5.122

7202.010219.4153.0

PrRe153.03

1

31

638.04

638.0

==

=Nu



90/174

4:

(+'ample 3.3)(+'ample 3.3)


#he surface area of the he'agon is%#he surface area of the he'agon is%

( ) Cm(Cm

(

m

NuD

kh

==

=

24.345.12210.0

02808.0

( )

( ) ( )

( )2

346.0

60sin

110.03

60sin26

m

mm

L

D

As

=

=

=

2*:U


91/174

+'ample 1+'ample 1


92/174

+'ample 1+'ample 1

42

+'ample 2+'ample 2


93/174

+'ample 2+'ample 2

43

+'ample 3+'ample 3


94/174

+'ample 3+'ample 3

4


95/174

4"

3.3. !rinciple of dynamic similarity!rinciple of dynamic similarity

and dimensional analysisand dimensional analysis(applied to forced convection)(applied to forced convection)

&on6dimensionali8ed&on6dimensionali8ed


96/174

4*

convection equationsconvection equations

#he continuity , momentum, and energy equations for steady,#he continuity , momentum, and energy equations for steady,incompressible, laminar flo$ of a fluid $ith constant propertiesincompressible, laminar flo$ of a fluid $ith constant properties

can be non6dimensionali8ed by dividing all the dependent andcan be non6dimensionali8ed by dividing all the dependent and

independent variables, as follo$s%independent variables, as follo$s%

&ote% the asteris;s denote non6dimensional variables.&ote% the asteris;s denote non6dimensional variables.

s

s

TT

TTT

V

##

V

vv

V

uu

Lyy

L!!

=

=

==

==

*

2

*

**

**

;

;

;;

Free stream velocity

Free stream temperature


97/174

40


ntroducing these variables the equations become%ntroducing these variables the equations become%

Continuity:Continuity:

Momentum:Momentum:

Energy:Energy:

0*

*

*

*

=

+

y

v

!

u

*

*

2*

*2

*

**

*

**

Re

1

d!

d#

y

u

y

uv

!

uu

L

=

+

2*

2

*

**

*

**

PrRe

1

y

T

y

Tv

!

Tu

L

=

+

&on6dimensionali8ed&on6dimensionali8ed


98/174

4


For a plate, the boundary conditions are%For a plate, the boundary conditions are%

( ) ( ) ( )( ) ( )

( ) ( ) 1,1,00,00,

1,000,1,0

****

****

******

==

==

===

!T!u

!T!u

yT!vyu

'V'V

yVyV

##ss

uu

, #, #


99/174

44


100/174

1::


101/174

1:1


102/174

1:2


103/174

1:3


104/174


105/174

1:"

3."3." eynolds Analogyeynolds Analogy



106/174

1:*

(rag Force)(rag Force)

Forced ConvectionForced Convection( ld A l )


107/174

1:0

(eynolds Analogy)(eynolds Analogy)

n forced convection analysis, $e are primarilyn forced convection analysis, $e are primarilyinterested in the determination of quantities of%interested in the determination of quantities of%

#he coefficient of friction (C#he coefficient of friction (CFF) (to calculate the) (to calculate the

shear stress at the $all)shear stress at the $all) &usselt number (&u) ( to calculate the heat&usselt number (&u) ( to calculate the heat

transfer rates).transfer rates).

#herefore, it is desirable to have a relation bet$een#herefore, it is desirable to have a relation bet$eenCCFFand &u, so that $e can calculate one $hen theand &u, so that $e can calculate one $hen the

other is available.other is available.

Forced ConvectionForced Convection( ld A l )


108/174

1:



109/174

1:4


( )

( )L

L

L

y

V

yy

uL

V

Vs!f

!f

!f

y

uC

Re,

Re,

Re

2

Re

2

*

3

*

2

0*

*

*

2

0*

2

, 2

*

*

2

=

=

=

===

=


110/174

11:



111/174

111



112/174

112



( )

( )

0*

*

*

0*

*

*

=

=

=

=

y

ys

s

y

T

L

ky

T

TT!

TTkh

Forced ConvectionForced Convection( ld A l )( ld A l )


113/174

113


114/174

11


&ote% the &usselt number is equivalent to the&ote% the &usselt number is equivalent to thedimensionless temperature gradient at the surface, anddimensionless temperature gradient at the surface, and

this is $hy it is sometimes called the dimensionless heatthis is $hy it is sometimes called the dimensionless heat

transfer coefficient (h).transfer coefficient (h).

Fig 6!"# (text)

Forced ConvectionForced Convection( ld A l )(eynolds Analogy)


115/174

11"


#he average friction and heat transfer coefficients#he average friction and heat transfer coefficientsare determined by integrating the local Care determined by integrating the local CF,'F,'and &uand &u''

over the surface of the given body $ith respect to 'Vover the surface of the given body $ith respect to 'V

(from : to :.1), $hich removes the dependence on 'V(from : to :.1), $hich removes the dependence on 'V

and thus gives%and thus gives%

#hese relations allo$ e'perimenters to study a#hese relations allo$ e'perimenters to study aproblem $ith a minimum amount of e'periments andproblem $ith a minimum amount of e'periments and

report their results in terms of Dust e and !r.report their results in terms of Dust e and !r.

( ) ( )Pr,ReRe 34 LLF %NuandfC ==

Forced ConvectionForced Convection( ld A l )(eynolds Analogy)


116/174

11*


#he e'perimental data for heat transfer is often#he e'perimental data for heat transfer is oftenrepresented ($ith reasonable accuracy) by a simplerepresented ($ith reasonable accuracy) by a simple

po$er la$ relation of the form%po$er la$ relation of the form%

5here m and n are constant e'ponents (normally bet$een :5here m and n are constant e'ponents (normally bet$een :

and 1), and the value of C depends on geometry.and 1), and the value of C depends on geometry.

nmLCNu PrRe =

Forced ConvectionForced Convection(eynolds Analogy)(eynolds Analogy)


117/174

110



118/174

11


&o$ if $e simplify the momentum and energy&o$ if $e simplify the momentum and energyequations by assuming%equations by assuming%

!r > 1 ($hich is appro'imately true for gases)!r > 1 ($hich is appro'imately true for gases)

(true $hen u > u(true $hen u > u

> ?> ?

> constant)> constant)

0*

*

=!

#

For !r > 1, theFor !r > 1, the

thermal andthermal andvelocity boundaryvelocity boundary

layers coincidelayers coincide



119/174

114

(eynolds Analogy)(eynold s Analogy)

#he equations then become%#he equations then become%

Momentum:Momentum:

Energy:Energy:

&ote% #hese t$o equations are e'actly in the same&ote% #hese t$o equations are e'actly in the same

form for uV and #V.form for uV and #V.

2*

*2

*

**

*

**

2*

*2

*

**

*

**

Re

1

Re

1

y

T

y

Tv

!

Tu

y

u

y

uv

!

uu

L

L

=

+

=

+



120/174

12:


121/174

121

(eynold s Analogy)(eynold s Analogy)


122/174

122


#herefore substituting these values into#herefore substituting these values into +quation+quationVVgives%gives%

0*

*

*

0*

*

*

==

=

yy y

T

y

u

2

2

Re

,

,

!F

!

!F

!

C)t

or

CNu

=

= eynolds Analogy for

!r > 1

Forced ConvectionForced Convection(


123/174

123

(


124/174

12


eynolds Analogy is important because it allo$s useynolds Analogy is important because it allo$s us

to determine the heat transfer coefficient (h) for fluidsto determine the heat transfer coefficient (h) for fluids

$here !r > 1, from ;no$ledge of the friction$here !r > 1, from ;no$ledge of the friction

coefficient ($hich is easier to measure).coefficient ($hich is easier to measure).

Forced ConvectionForced Convection(Chilton6Colburn Analogy)(Chilton6Colburn Analogy)


125/174

12"

(Chilton6Colburn Analogy)(Chilton6Colburn Analogy)

Ho$ever, the eynolds number is of limited useHo$ever, the eynolds number is of limited usebecause of the restrictions%because of the restrictions%

!r > 1!r > 1

#herefore it is desirable to have an analogy that is#herefore it is desirable to have an analogy that isapplicable over a $ide range of !r.applicable over a $ide range of !r.

#his is done by adding a#his is done by adding a !randtl number correction!randtl number correction..

0*

*

=!

#



126/174

12*


ecall as previously derived%ecall as previously derived%

#a;ing their ratio and rearranging give the relation#a;ing their ratio and rearranging give the relation

;no$n as the;no$n as the Chilton6Colburn analogyChilton6Colburn analogyor theor the

modified eynoldWs analogymodified eynoldWs analogy%%

21

31

21

RePr332.0Re664.0, !!!!F NuandC ==

HL!

!F*Nu

C== 1, RePr

2

31

32

Pr2

,

==VC

hC*

p

!!F

H

For :.* R !r R *:

Colburn D6factorColburn D6factor



127/174

120


#he Chilton6Colburn Analogy is derived using%#he Chilton6Colburn Analogy is derived using% /aminar flo$/aminar flo$

Gver a flat plate ( )Gver a flat plate ( )

Ho$ever, e'perimental studies ho$ever sho$ that it is alsoHo$ever, e'perimental studies ho$ever sho$ that it is also

appro'imately applicable toappro'imately applicable to turbulent flo$turbulent flo$over a surface inover a surface in

thethe presence of pressure gradientspresence of pressure gradients..

For laminar flo$ it isFor laminar flo$ it is notnotapplicable unless it is a flat plate,applicable unless it is a flat plate,

therefore it cannot be applied to laminar flo$ in a pipe.therefore it cannot be applied to laminar flo$ in a pipe.

Also the analogy above can be used forAlso the analogy above can be used for locallocaloror averageaverage

quantities.quantities.

0=!

#

Forced ConvectionForced Convection(+'ample 3 )(+'ample 3 )


128/174

12

(+'ample 3.)(+'ample 3.)

+'ample 3.+'ample 3. /aminar flo$ profile/aminar flo$ profileover a vertical plate.over a vertical plate. A 2 ' 3 m plateA 2 ' 3 m plate

is suspended in a room and subDectis suspended in a room and subDect

to air flo$ parallel to its surfacesto air flo$ parallel to its surfaces

along its 3 m side. #he total dragalong its 3 m side. #he total dragforce acting on the plate is :.* &.force acting on the plate is :.* &.

etermine the average heat transferetermine the average heat transfer

coefficient (h) for the plate%coefficient (h) for the plate%

#he properties of air at 1 atm (#able A61" in#he properties of air at 1 atm (#able A61" in

te't boo;) at #te't boo;) at #filmfilm> 2:> 2:C%C%

3 m

2 m

Air Flo$Air Flo$##

> 1"> 1"CC

??

> 0 ms> 0 ms

7309.0Pr

007.1;204.13

=

==k%

k+C

m

k%p

#s>2"C

Forced ConvectionForced Convection(+'ample 3 )(+'ample 3 )


129/174

124


3 m J Characteristic length 3 m J Characteristic length


130/174

13:

For all flat plates%For all flat plates%

Drag = Friction ForceDrag = Friction Force



( )

( ) ( ) ( ) 00243.0

712204.1

86.022222

3

=

=

= s

m

m

k%s

Fm

N

VA

DC

221

== VACDF sFfriction

Forced ConvectionForced Convection(+'ample 3 )(+'ample 3.)


131/174

131


#hen from the modified eynolds analogy (Chilton6#hen from the modified eynolds analogy (Chilton6Colburn) the average heat transfer coefficient (h) canColburn) the average heat transfer coefficient (h) can

be calculated%be calculated%

( ) ( ) ( )

Cm

(

Ck%+

sm

m

k%

pF CVCh

=

=

=

2

32

3

32

7.127309.0

10077204.1

2

00243.0

Pr2


132/174

132

3.*3.* Convection in anConvection in an

internal flo$internal flo$


133/174

nternal Flo$nternal Flo$(&on6Circular #ubes)(&on6Circular #ubes)


134/174

13

(&on Circular #ubes)( o C cu a ubes)

For flo$ through non6For flo$ through non6circular tubes e and &u,circular tubes e and &u,

are based on the hydraulicare based on the hydraulic

diameter diameter hh..

5here p is the perimeter, ?5here p is the perimeter, ?mmisis

the mean velocity, and Athe mean velocity, and Accis theis the

cross6sectional area.cross6sectional area.

hm

ch

DV

pAD

=

=

Re

4

nternal Flo$nternal Flo$(Iean ?elocity)(Iean ?elocity)


135/174

13"

(Iean ?elocity)( y)

ecause the velocity varies over the cross6section itecause the velocity varies over the cross6section itis necessary to $or; $ith a mean velocity (?is necessary to $or; $ith a mean velocity (?mm) $hen) $hen

dealing $ith internal flo$s.dealing $ith internal flo$s.

c

m

mc

A

mV

VAm

=

=

nternal Flo$nternal Flo$(Circular #ubes)(Circular #ubes)


136/174

13*

( )( )

n a circular tube%n a circular tube%

e R 2,3::e R 2,3:: laminar flo$laminar flo$ 2,3:: R e R 1:,:::2,3:: R e R 1:,::: transitional flo$transitional flo$

e O 1:,:::e O 1:,::: turbulent flo$turbulent flo$

DV

DDp

AD

m

D

ch

=

=

=

=

Re

44 42

nternal Flo$nternal Flo$(+ntrance egion)(+ntrance egion)


137/174

130

( g )( g )

nternal Flo$ +quationsnternal Flo$ +quations(+'ample 3.")(+'ample 3.")


138/174

13

( a p e 3 ")( p )

+'ample 3." 6+'ample 3." 6 #emperature rise of oil in a bearing#emperature rise of oil in a bearing(a)(a) Find the temperature and velocity distributionsFind the temperature and velocity distributions

(b)(b) Find the ma'imum temperature in the oilFind the ma'imum temperature in the oil

(c)(c) Find the ma'imum heat flu' in the oilFind the ma'imum heat flu' in the oil

u(y)u(y)/> 2 mm

?> 12 ms?> 12 mspper plate movingpper plate moving

Gil

;> :.1" 5(m@)

X> :. ;g(ms)

/o$er plate stationary/o$er plate stationary



139/174

134

( p )( p )

Assumptions%Assumptions% 12 ms?> 12 mspper plate movingpper plate moving

Gil;> :.1" 5(m@)

X> :. ;g(ms)

/o$er plate stationary/o$er plate stationary



140/174

1:

( p )( p )

(a) Find the temperature and velocity distributions :

Continuity +quation%Continuity +quation%

#he '6component of velocity does not change.


141/174

11

( p )( p )

'6momentum equation%'6momentum equation%

#his is a 2#his is a 2ndndorder differential equation.


142/174

12

( p )( p )

#he boundary conditions are%#he boundary conditions are%

u(:)> :u(:)> :

u(/)> ?> 12 msu(/)> ?> 12 ms

sing these boundary conditions to solve for the constants Csing these boundary conditions to solve for the constants C11and Cand C22gives%gives%

( )

0

00

2

21

=+=

C

CC ( )

L

V

C

LCV

=

+=

1

1 0



143/174

13

( p )( p )

#herefore the equation becomes%#herefore the equation becomes%

Frictional heating due to viscous dissipation in this case isFrictional heating due to viscous dissipation in this case is

significant because of the high viscosity of oil and large platesignificant because of the high viscosity of oil and large plate

velocity. #he plates are isothermal and there is no change invelocity. #he plates are isothermal and there is no change in

flo$ direction, so the temperature changes $ith y only #> #(y).flo$ direction, so the temperature changes $ith y only #> #(y).

VL

yu =



144/174

1

:: ::

:: :: ::

( p )( p )


145/174

1"

( p )

2

2

2

=

LV

yTk


146/174

1*

( )

&o$ integrating the equation t$ice%&o$ integrating the equation t$ice%

Applying boundary conditions%Applying boundary conditions%

#(:) > ##(:) > #::

#(/) > ##(/) > #::

43

2

2CyCV

L

y

kT ++

=

40:0 CTy ==

2

3

03

2

0

2

2:

VkL

C

TLCVL

L

kTLy

=

++

==


147/174



148/174

1

(b) Find the ma'imum temperature in the oil#he temperature gradient is found by differentiating #(y) $ith

respect to y.

&o$ to find the ma'imum temperature, ma'imi8e # by setting

the above equation equal to :.

021

2

2

=

=

L

y

kL

V

y

T

mmL

y

L

y

001.02

002.0

2

21

====


149/174



150/174

1":

(c) Find the ma'imum heat flu' in the oil#he heat flu' at the plates is determined from the definition of a

heat flu'.

( ) ( )

( )

( )

( )

2

22

2

0

0

800,28

1

1

002.02

128.0

2

212

2

m

(

(

mL

V

L

y

kL

V

kdy

dT

kq

s

mN

sm

m

sN

y

=

=

=

==

=



151/174

1"1

As a chec;, $e can also calculate the heat flu' at y> / (shouldAs a chec;, $e can also calculate the heat flu' at y> / (shouldbe equal but opposite sign).be equal but opposite sign).

( ) ( )( )

( )

( )

2

22

2

800,28

1

1

002.02

128.0

2

21

2

2

m

(

(

mL

V

L

L

kL

Vk

dy

dTkq

smN

sm

m

sN

Ly

L

+=

+=

+=

==

=

Correct Y



152/174

1"2

iscussion of e'ampleiscussion of e'ample

A temperature rise of 44A temperature rise of 44QC confirms that viscous dissipation isQC confirms that viscous dissipation is

very significantvery significant

#>114#>114CC/> 2 mm

?> 12 ms?> 12 mspper plate movingpper plate moving

/o$er plate stationary/o$er plate stationary#>2:#>2:CC

#>2:#>2:CC


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3.03.0 Free (natural) convectionFree (natural) convection


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Free ConvectionFree Convection(?olume +'pansion Coefficient)(?olume +'pansion Coefficient)


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n heat transfer, the primary variable is then heat transfer, the primary variable is thetemperature, so it is desirable to e'press the nettemperature, so it is desirable to e'press the net

buoyancy force in terms of a temperature difference.buoyancy force in terms of a temperature difference.

#his requires ;no$ledge of a property that represents the#his requires ;no$ledge of a property that represents the

variation of the density of a fluid $ith temperature at constantvariation of the density of a fluid $ith temperature at constantpressure.pressure.

#his is called the volume e'pansion coefficient (#his is called the volume e'pansion coefficient (ZZ) $hich is) $hich is

defined as%defined as%

## TT

=

=

11



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n natural convection studies, the condition of the fluidn natural convection studies, the condition of the fluidsufficiently far from the hot or cold surface is indicated by thesufficiently far from the hot or cold surface is indicated by the

subscript [subscript [

\ to indicate that the presence of the surface is not\ to indicate that the presence of the surface is not

felt.felt.

n such cases,n such cases, Z can be e'pressed appro'imately by replacingZ can be e'pressed appro'imately by replacing

the differential equations by differences, such as%the differential equations by differences, such as%

( )

( )TTT =

=

11

( ) = TT



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For an ideal gas%For an ideal gas%

#hus for an#hus for an ideal gasideal gasthe discharge coefficientthe discharge coefficientbecomes%becomes%

T,

#

=

TTT,T#

,T#

#

111

=

=

=

Free ConvectionFree Convection(7rashof &umber)(7rashof &umber)


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#he velocity and temperature for natural#he velocity and temperature for naturalconvectionconvection over a vertical plateover a vertical plateareare

sho$n in the figure.sho$n in the figure.

As in forced convection, the boundaryAs in forced convection, the boundary

layer thic;ness increases in the flo$layer thic;ness increases in the flo$

directiondirection

nli;e forced convection, the fluidnli;e forced convection, the fluid

velocity (u) is : at the outer edge ofvelocity (u) is : at the outer edge of

the boundary layer as $ell as thethe boundary layer as $ell as the

surface of the plate.surface of the plate.

#his is e'pected since the fluid#his is e'pected since the fluid

beyond the boundary layer isbeyond the boundary layer is

motionless.motionless.



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ecall that the '6momentum equations is%ecall that the '6momentum equations is%

&o$ the momentum equation outside the boundary layer can be&o$ the momentum equation outside the boundary layer can be

obtained from this relation as a special case by setting u > :,obtained from this relation as a special case by setting u > :,

giving%giving%

%!

#

y

u

y

uv

!

uu

=

+

2

2

%!

#=



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1*1


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1*2

f $e no$ non6dimensionali8e this '6momentumf $e no$ non6dimensionali8e this '6momentumequation, $e get%equation, $e get%

( )2*

*2

2

*

2

3

*

**

*

**

Re

1

Re y

uTLTT%

y

u

v!

u

u LL

cs

+

=

+

7rashof &umber7rashof &umber



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1*3

#he 7rashof number is derived by#he 7rashof number is derived byFran8 7rashof (12*6143) fromFran8 7rashof (12*6143) from

7ermany.7ermany.

( )2

3

csL

LTT%-r =



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7r is a measure of the relative7r is a measure of the relativemagnitudes of the buoyancy forcemagnitudes of the buoyancy force

and the opposing viscous forceand the opposing viscous force

acting on the fluidacting on the fluid

Free ConvectionFree Convection(aleigh &umber)(aleigh &umber)


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/ord aleigh (1261414) from/ord aleigh (1261414) from+ngland derived the aleigh &umber+ngland derived the aleigh &umber

Pr=-r,a

( )Pr

2

3

=

csL

LTT%,a

Free ConvectionFree Convection(+'ample 3.*)(+'ample 3.*)


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1**

+'ample 3.*+'ample 3.* A *6m long section of 6cm diameterA *6m long section of 6cm diameterhori8ontal hot $ater pipe passes through a largehori8ontal hot $ater pipe passes through a large

room. #he pipe surface temperature is 0:room. #he pipe surface temperature is 0: QC.QC.

etermine the heat loss from the pipe by naturaletermine the heat loss from the pipe by natural

convection.convection.

> cm> cm

/> * m/> * m

##ss> 0:> 0: QCQC##

> 2:> 2: QCQC



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1*0

Assume%Assume%


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1*

#he volumetric e'pansion coefficient (#he volumetric e'pansion coefficient (ZZ) is%) is%

#he characteristic length is the outer diameter of the#he characteristic length is the outer diameter of the

pipe%pipe%

.CTf 318

1

27345

11=

+==

mDLc 08.0==



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1*4

#herefore the aleigh &umber is%#herefore the aleigh &umber is%

( )

( ) ( ) [ ] ( ) ( )( )

6

25

3

3181

2

3

10869.1

10749.1

7241.008.029334381.9

Pr

2

2

=

=

=

sm

.s

m

sD

m..

DTT%,a



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10:

#able 461 in the te't boo; gives average &usselt#able 461 in the te't boo; gives average &usseltnumbers for natural convection over surfaces.numbers for natural convection over surfaces.

For a hori8ontal cylinder%For a hori8ontal cylinder%



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101

#hus &u is%#hus &u is%

( ) 4.17

7241.0

559.01

10869.1387.060.0

Pr

559.0

1

387.060.0

2

6

2

278

169

61

278

169

61

=

+

+=

+

+= DD

,aNu



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102

#hen%#hen%

#he surface area of the cylinder is%#he surface area of the cylinder is%

( )

( ) ( )

Cm

(Cm(

mNu

D

kh

=== 2869.54.17

08.0

02699.0

( ) ( ) 2508.1608.0 mmm

LDAs

== =



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103

#herefore the heat transfer is%#herefore the heat transfer is%

( )

( ) ( ) ( )(

CCm

TTAhQ

Cm(

ss

5.442

2070508.1869.5 2

=

=

=

EEnd Ofnd Of CConvectiononvection SSectionectionC C


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Mec 551 Convection