Lect - 11 Internal Forced convection.pptx

8/17/2019 Lect - 11 Internal Forced convection.pptx

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Internal Forced convectionDr. Senthilmurugan S. Department of Chemical Engineering IIT Guwahati - Part 11


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5/12/16 | Slide 2

Objectives

Obtain average velocity from a knowledge of velocity

rofile! and average temerat"re from a knowledge of

temerat"re rofile in internal flow

#ave a vis"al "nderstanding of different flow regions in

internal flow! and calc"late $ydrodynamic and t$ermal entry

lengt$s %naly&e $eating and cooling of a fl"id flowing in a t"be

"nder constant s"rface temerat"re and constant s"rface

$eat fl"' conditions! and work wit$ t$e logarit$mic mean

temerat"re difference

Obtain analytic relations for t$e velocity rofile! ress"re

dro! friction factor! and ("sselt n"mber in f"lly develoed

laminar flow

)etermine t$e friction factor and ("sselt n"mber in f"lly

develoed t"rb"lent flow "sing emirical relations! and

calc"late t$e $eat transfer rate


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5/12/16 | Slide *

Introd"ction

+i,"id or gas flow t$ro"g$ ies or d"cts is commonly

"sed in $eating and cooling alications and fl"id

distrib"tion networks-

.$e fl"id in s"c$ alications is "s"ally forced to flow by

a fan or "m t$ro"g$ a flow section-

%lt$o"g$ t$e t$eory of fl"id flow is reasonably well"nderstood! t$eoretical sol"tions are obtained only for a

few simle cases s"c$ as f"lly develoed laminar flow in

a circ"lar ie-

.$erefore! we m"st rely on e'erimental res"lts and

emirical relations for most fl"id flow roblems rat$er

t$an closedform analytical sol"tions-

For a fi'ed s"rface area! t$e circ"lar t"be gives t$e most

$eat transfer for t$e least ress"re dro-

0irc"lar ies can wit$stand large ress"re differences

between t$e inside and t$e o"tside wit$o"t "ndergoing

any significant distortion! b"t noncirc"lar ies cannot-


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5/12/16 | Slide

Flow t$ro"g$ t"bes

.$e fl"id velocity in a ie c$anges from

&ero at t$e wall beca"se of t$e nosli

condition to a ma'im"m at t$e ie

center-

In fl"id flow! it is convenient to work wit$

an average velocity avg! w$ic$remains constant in incomressible flow

w$en t$e crosssectional area of t$e

ie is constant-

.$e average velocity in $eating and

cooling alications may c$ange

somew$at beca"se of c$anges indensity wit$ temerat"re-

3"t! in ractice! we eval"ate t$e fl"id

roerties at some average temerat"re

and treat t$em as constants-

Average velocity V avg is defned asthe average speed through a crosssection.

For ully developed laminar pipeow, V avg is hal o the maximum

velocity.


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5/12/16 | Slide 5

%verage velocity

.$e val"e of t$e average 4mean velocity V avg at some streamwise crosssection

$ere w$ere m is t$e mass flow rate! r is t$e density! Ac is t$e crosssectional area!

and u4r is t$e velocity rofile-

.$e average velocity for incomressible flow in a circ"lar ie of radi"s R

.$erefore! w$en we know t$e flow rate or t$e velocity rofile! t$e average velocity

can be determined easily-

Internal Forced Flow


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5/12/16 | Slide 6

%verage .emerat"re

In fl"id flow! it is convenient to work wit$ an

average or mean temerat"re .m! w$ic$

remains constant at a cross section- .$e mean

temerat"re .m c$anges in t$e flow direction

w$enever t$e fl"id is $eated or cooled-

(ote t$at t$e mean temerat"re .m of a fl"idc$anges d"ring $eating or cooling- %lso! t$e fl"id

roerties in internal flow are "s"ally eval"ated

at t$e b"lk mean fl"id temerat"re! w$ic$ is t$e

arit$metic average of t$e mean temerat"res at

t$e inlet and t$e e'it- .$at is! .b 7 4.i 8.e/2-

Internal Forced Flow

Actual and idealized temperatureprofles or ow in a tube (the rateat which energy is transported withthe uid is the same or both cases.


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5/12/16 | Slide 9

+aminar and ."rb"lent Flow in ."bes

Flow in a t"be can be laminar or t"rb"lent! deending

on t$e flow conditions-

Fl"id flow is streamlined and t$"s laminar at low

velocities! b"t t"rns t"rb"lent as t$e velocity is

increased beyond a critical val"e-

.ransition from laminar to t"rb"lent flow does not

occ"r s"ddenly: rat$er! it occ"rs over some range of

velocity w$ere t$e flow fl"ct"ates between laminarand t"rb"lent flows before it becomes f"lly t"rb"lent-

;ost ie flows enco"ntered in ractice are t"rb"lent-

+aminar flow is enco"ntered w$en $ig$ly visco"s

fl"ids s"c$ as oils flow in small diameter t"bes or

narrow assages-

.ransition from laminar to t"rb"lent flow deends on

t$e ! f"lly

t"rb"lent for !>>>! and transitional in between-


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5/12/16 | Slide @

.$e $ydra"lic diameter

A For flow t$ro"g$ noncirc"lar

t"bes! t$e


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5/12/16 | Slide C

.$e entrance region

elocity bo"ndary layer 4bo"ndary layer B .$e

region of t$e flow in w$ic$ t$e effects of t$e visco"s

s$earing forces ca"sed by fl"id viscosity are felt-

.$e $yot$etical bo"ndary s"rface divides t$e flow

in a ie into two regionsB

3o"ndary layer regionB .$e visco"s effects and t$e

velocity c$anges are significant-

Irrotational 4core flow regionB .$e frictional effects

are negligible and t$e velocity remains essentially

constant in t$e radial direction-

#ydrodynamic entrance regionB .$e region from

t$e ie inlet to t$e oint at w$ic$ t$e velocity

rofile is f"lly develoed-

#ydrodynamic entry lengt$ LhB .$e lengt$ of t$is

region-

#ydrodynamically f"lly develoed regionB .$e

region beyond t$e entrance region in w$ic$ t$e

velocity rofile is f"lly develoed and remains

"nc$anged-

elocity bo"ndary layer

!he development o the velocityboundary layer in a pipe. (!hedeveloped average velocity profleis parabolic in laminar ow, asshown, but much atter or uller inturbulent ow.

Flow in t$e entrance region is called

hydrodynamically developing flow since t$is is

t$e region w$ere t$e velocity rofile develos-


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5/12/16 | Slide 1>

.$e entrance region

.$ermal entrance regionB .$e region of

flow over w$ic$ t$e t$ermal bo"ndary

layer develos and reac$es t$e t"be

center- .$ermal entry lengt$B .$e lengt$ of t$is

region- .$ermally develoing flowB Flow in t$e

t$ermal entrance region- .$is is t$e

region w$ere t$e temerat"re rofile

develos- .$ermally f"lly develoed regionB .$e

region beyond t$e t$ermal entranceregion in w$ic$ t$e dimensionless

temerat"re rofile remains "nc$anged- F"lly develoed flowB .$e region in w$ic$

t$e flow is bot$ $ydrodynamically and

t$ermally develoed-

.$ermal bo"ndary layer

.$e fl"id roerties in internal flow are

"s"ally eval"ated at t$e bulk mean fluidtemperature! w$ic$ is t$e arit$metic

average of t$e mean temerat"res at t$e

inlet and t$e e'itB T b = 4T m, i + T m, e/2

.$e develoment of t$e t$ermal bo"ndary

layer in a t"be


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5/12/16 | Slide 11

)efinition for elocity and t$ermal bo"ndary layer

#ydrodynamically f"lly develoedB

.$ermally f"lly develoed

In t$e t$ermally f"lly develoed region of a t"be! t$e

local convection coefficient is constant 4does not vary

wit$ '-

.$erefore! bot$ t$e friction 4w$ic$ is related to wall

s$ear stress and convection coefficients remainconstant in t$e f"lly develoed region of a t"be-

.$e ress"re dro and $eat fl"' are $ig$er in t$e

entrance regions of a t"be! and t$e effect of t$e

entrance region is always to increase t$e average

friction factor and $eat transfer coefficient for t$e

entire t"be-


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5/12/16 | Slide 12

Dntry +engt$s

.$e $ydrodynamic entry lengt$ is

"s"ally taken to be t$e distance from t$e

t"be entrance w$ere t$e wall s$ear

stress 4and t$"s t$e friction factor

reac$es wit$in abo"t 2 ercent of t$e

f"lly develoed val"e- In laminar flow! t$e $ydrodynamic and

t$ermal entry lengt$s are given

aro'imately as Esee ays and

0rawford 41CC* and S$a$ and 3$atti

41C@9G

In t"rb"lent flow! t$e intense mi'ing

d"ring random fl"ct"ations "s"ally

overs$adows t$e effects of molec"lar

diff"sion! and t$erefore t$e

$ydrodynamic and t$ermal entry lengt$s

are of abo"t t$e same si&e andindeendent of t$e Hrandtl n"mber-

.$e entry lengt$ is m"c$ s$orter in

t"rb"lent flow

Its deendence on t$e


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5/12/16 | Slide 1

Internal forced convection $eat transfer 0onstant s"rface $eat fl"' and temerat"re

0onstant s"rface $eat fl"'0onstant s"rface temerat"re

.$e constant s"rface temerat"re condition

is reali&ed w$en a $ase c$ange rocess

s"c$ as boiling or condensation occ"rs at

t$e o"ter s"rface of a t"be-

.$e constant s"rface $eat fl"' condition is

reali&ed w$en t$e t"be is s"bjected to

radiation or electric resistance $eating

"niformly from all directions-

e may $ave eit$er T s = constant or s = constant att$e s"rface of a t"be! b"t not bot$-


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5/12/16 | Slide 15

0onstant S"rface #eat Fl"' 4,/% 7 constant

S"rface $eat fl"' is e'ressed as

.$e rate of $eat transfer can also be

e'ressed as

;ean fl"id temerat"re at t$e t"be e'it

S"rface temerat"re

For f"lly develoed flow $ 7 constant$ the uid properties remain constant

during ow


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5/12/16 | Slide 16

0onstant S"rface #eat Fl"' 4,/% 7 constant

Dnergy balance steadyflow energy

balance

For constant s"rface $eat fl"' condition

$en bot$ s !A and h are constant t$en

$en '7+ .m7.e


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5/12/16 | Slide 19

0onstant s"rface temerat"re

From (ewtons law of cooling! t$e rate of

$eat transfer to or from a fl"id flowing in a

t"be can be e'ressed as

w$ere $ is t$e average convection $eattransfer coefficient! %s is t$e $eat transfer

s"rface area-

∆.avg is some aroriate average

temerat"re difference between t$e fl"id

and t$e s"rface-

.wo s"itable ways of e'ressing ∆.avg %rit$metic mean temerat"re

difference

+ogarit$mic mean temerat"re

difference

%rit$metic mean temerat"re difference

.b is t$e b"lk mean fl"id temerat"re!

3y "sing arit$metic mean temerat"re

difference! we ass"me t$at t$e mean fl"id

temerat"re varies linearly along t$e t"be!

w$ic$ is $ardly ever t$e case w$en .s 7

constant- .$is simle aro'imation often gives

accetable res"lts! b"t not always-

.$erefore! we need a better way to

eval"ate ∆.avg-

Internal forced convection in t"bes


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5/12/16 | Slide 1@

0onstant s"rface temerat"re

.$e energy balance on a differential

control vol"me

Integrating from " 7 > 4t"be inlet w$ere

Tm = Ti to " 7 L 4t"be e'it w$ere Tm 7

Te gives

Internal forced convection in t"bes

d'


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5/12/16 | Slide 1C

Stream energy 3alance

3y combining bot$ e,"ation

0onstant s"rface temerat"reInternal forced convection in t"bes


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5/12/16 | Slide 2>

Flow .$o"g$ ."bes J +aminar flow

%ly t$e energy balance on a differential

vol"me element! and solve it to obtain t$e

temerat"re rofile for t$e constant s"rface

temerat"re and t$e constant s"rface $eat

fl"' cases


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5/12/16 | Slide 21

.emerat"re rofile and t$e ("sselt ("mber

.$e steadyflow energy balance for a

cylindrical s$ell element of t$ickness dr

and lengt$ d" can be e'ressed as

S"bstit"ting and dividing by 2πrdrd"gives! after rearranging

3"t from Fo"riers law of $eat

cond"ction in t$e radial direction

Dnergy balance J flow t$ro"g$ ie laminar

%ubstituting and using α & k 'ρc p gives


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5/12/16 | Slide 22

.emerat"re rofile and t$e ("sselt ("mber

For +aminar f"lly develoed flow!

constant $eat fl"'

If $eat cond"ction in t$e " direction were

considered in t$e derivation of Dnergy

balance! it wo"ld give an additional term !

w$ic$ wo"ld be e,"al to &ero since

constant and t$"s T = T 4r - .$erefore! t$e

ass"mtion t$at t$ere is no a'ial $eat

cond"ction is satisfied e'actly in t$is

case-

.$e relation for laminar velocity rofile

w$ic$ is a secondorder ordinary

differential e,"ation- Its general sol"tion

is obtained by searating t$e variables

and integrating twice to be

0onstant S"rface #eat Fl"'


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5/12/16 | Slide 2*

F"lly develoed laminar flow in a circ"lar t"be-

Sol"tion for energy balance e,"ation

301B at r 7 >

302B . 7 .s at r 7 <

.$e mean temerat"re .m is

determined by s"bstit"ting t$e velocity

and temerat"re rofile relations

s"bjected to constant s"rface $eat fl"'! t$e ("sselt n"mber is a constant- .$ere is no

deendence on t$e


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5/12/16 | Slide 2

F"lly develoed laminar flow in a circ"lar t"be

% similar analysis can be erformed for

f"lly develoed laminar flow in a circ"lar

t"be for t$e case of constant s"rface

temerat"re .s- .$e sol"tion roced"re in

t$is case is more comle' as it re,"ires

iterations! b"t t$e ("sselt n"mber relation

obtained is e,"ally simle

.$e ("sselt n"mber for t$e case of

constant s"rface $eat fl"' is 16 ercent

$ig$er t$an t$e case of constant s"rfacetemerat"re for t$e f"lly develoed

laminar ie flow-

.$is is contrary to t$e res"lts t$e t"rb"lent

flow 4s$own earlier slide 1*

.$is s$ows t$at laminar flow is sensitive

to t$e alied s"rface t$ermal bo"ndary

condition and for alications re,"iring

$ig$er rates of $eat transfer! w$enever

ossible: t$e constant s"rface $eat fl"'

bo"ndary condition s$o"ld be "sed-

.$e t$ermal cond"ctivity k for "se in t$e

(" relations above s$o"ld be eval"ated

at t$e b"lk mean fl"id temerat"re!

w$ic$ is t$e arit$metic average of t$e

mean fl"id temerat"res at t$e inlet and

t$e e'it of t$e t"be- For laminar flow! t$e effect of s"rface

ro"g$ness on t$e friction factor and t$e

$eat transfer coefficient is negligible

For 0onstant S"rface .emerat"re


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5/12/16 | Slide 25

("sselt n"mber relati

.$e


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5/12/16 | Slide 26

)eveloing +aminar Flow in t$e Dntrance


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5/12/16 | Slide 29

Lraet& n"mber s ("sselt ("mber

.$e local val"es of ("sselt n"mber are

tyically resented eit$er gra$ically or

in tab"lar form in terms of t$e inverse of

a dimensionless arameter called t$e

Lraet& n"mber

w$ic$ is defined as L& 7 4$/ " 5


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5/12/16 | Slide 2@

."rb"lent Flow in ."bes

e mentioned earlier t$at flow in

smoot$ t"bes is "s"ally f"lly t"rb"lent

for !>>>-

."rb"lent flow is commonly "tili&ed in

ractice beca"se of t$e $ig$er $eat

transfer coefficients associated wit$ it- ;ost correlations for t$e friction and

$eat transfer coefficients in t"rb"lent

flow are based on e'erimental st"dies

beca"se of t$e diffic"lty in dealing wit$

t"rb"lent flow t$eoretically-

For smoot$ t"bes! t$e friction factor int"rb"lent flow can be determined from

t$e e'licit first Het"k$ov e,"ation

EHet"k$ov 41C9>G given as

.$e ("sselt n"mber in t"rb"lent flow is

related to t$e friction factor t$ro"g$ t$e

%hilton&%olburn analogy e'ressed as

Simplifed

Colburn equation

$ittus&'oelter euation

w$ere n = >- for heating and >-* for

cooling of t$e fl"id flowing t$ro"g$ t$e

t"be-

("sselt n"mber Dstimations


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5/12/16 | Slide 2C


w$en t$e temerat"re difference

between t$e fl"id and wall s"rface is not

large by eval"ating all fl"id roerties t

t$e b"lk mean fl"id temerat"re

.b 7 4.i 8 .e/2

$en t$e variation in roerties is larged"e to a large temerat"re difference!

t$e following e,"ation d"e to Sieder and

.ate 41C*6 can be "sedB

%ll roerties are eval"ated at T b e'cet

µs! w$ic$ is eval"ated at T s-

.$e ("sselt n"mber relations above are

fairly simle! b"t t$ey may give errors as

large as 25 ercent- .$is error can be

red"ced considerably to less t$an 1>ercent by "sing more comle' b"t

acc"rate relations s"c$ as t$e second

(etukhov euation e'ressed as

("sselt n"mber Dstimations


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5/12/16 | Slide *>


.$e acc"racy of t$is relation at lower


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5/12/16 | Slide *1


For li,"id metals 4>->> = Hr = >->1! t$e following relations are recommended bySleic$er and


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5/12/16 | Slide *2

F"lly )eveloed .ransitional Flow #eat .ransfer

.$e met$ods to $andle f"lly laminar and f"lly t"rb"lent $eat transfer $ave alreadybeen disc"ssed! $owever in some cases: t$e flow is in t$is transitional &one-

Fort"nately! t$e met$ods for $andling t"rb"lent flow can easily be adoted to deal

wit$ in t$is region-

% simle aroac$ is to contin"e to "se Lnielinskis 4 1C96 correlation along wit$ f

val"es determined from t$e following e'ressions for two common flow geometries


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5/12/16 | Slide **


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5/12/16 | Slide *


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5/12/16 | Slide *5


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5/12/16 | Slide *6


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5/12/16 | Slide *9

- %ct"al oerating conditions m"st

be considered in t$e design of iing systems-

%lso! t$e ;oody c$art and its e,"ivalent

0olebrook e,"ation involve several "ncertainties4t$e ro"g$ness si&e! e'erimental error! c"rve

fitting of data! etc-! and t$"s t$e res"lts obtained

s$o"ld not be treated as Me'act-N

.$ey are "s"ally considered to be acc"rate to 15

ercent over t$e entire range in t$e fig"re-

Friction factor


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5/12/16 | Slide *@

)eveloing ."rb"lent Flow in t$e Dntrance


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5/12/16 | Slide *C

."rb"lent Flow in (oncirc"lar ."bes

Hress"re dro and $eat transferc$aracteristics of t"rb"lent flow in t"bes

are dominated by t$e very t$in visco"s

s"blayer ne't to t$e wall s"rface! and

t$e s$ae of t$e core region is not of

m"c$ significance-

.$e t"rb"lent flow relations given above

for circ"lar t"bes can also be "sed for

noncirc"lar t"bes wit$ reasonable

acc"racy by relacing t$e diameter ) in

t$e eval"ation of t$e


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5/12/16 | Slide >

Flow t$ro"g$ ."be %nn"l"s

Some simle $eat transfer e,"iments consist oftwo concentric t"bes! and are roerly called

do"blet"be $eat e'c$angers

In s"c$ devices! one fl"id flows t$ro"g$ t$e t"be

w$ile t$e ot$er flows t$ro"g$ t$e ann"lar sace-

.$e governing differential e,"ations for bot$

flows are identical- .$erefore! steady laminar flow

t$ro"g$ an ann"l"s can be st"died analytically by

"sing s"itable bo"ndary conditions-

0onsider a concentric ann"l"s of inner diameter

$i and o"ter diameter $o- .$e $ydra"lic diameter

of t$e ann"l"s is

%nn"lar flow is associated wit$ two ("sselt

n"mbersP("i on t$e inner t"be s"rface and ("o on t$e o"ter t"be s"rfacePsince it may involve

$eat transfer on bot$ s"rfaces

.$e ("sselt n"mbers for f"lly develoed laminar

flow wit$ one s"rface isot$ermal and t$e ot$er

adiabatic are given in below table

F"lly develoed +aminar flow


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5/12/16 | Slide 1

Flow t$ro"g$ ."be %nn"l"s

For f"lly develoed t"rb"lent flow! $i and $o are aro'imately e,"al to eac$ ot$er!and t$e t"be ann"l"s can be treated as a noncirc"lar d"ct wit$ a $ydra"lic diameter

of )$ 7 )o Q )i-

.$e ("sselt n"mber can be determined from a s"itable t"rb"lent flow relation s"c$

as t$e Lnielinski e,"ation-

.o imrove t$e acc"racy! ("sselt n"mber can be m"ltilied by t$e followingcorrection factors w$en one of t$e t"be walls is adiabatic and $eat transfer is

t$ro"g$ t$e ot$er wallB

F"lly develoed t"rb"lent flow


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5/12/16 | Slide 2

#eat .ransfer Dn$ancement

."bes wit$ ro"g$ s"rfaces $ave m"c$ $ig$er$eat transfer coefficients t$an t"bes wit$

smoot$ s"rfaces-

.$erefore! t"be s"rfaces are often intentionally

ro"g$ened! corr"gated! or finned in order to

en$ance t$e convection $eat transfer

coefficient and t$"s t$e convection $eat

transfer rate-

#eat transfer in t"rb"lent flow in a t"be $as

been increased by as m"c$ as >> ercent by

ro"g$ening t$e s"rface-


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5/12/16 | Slide *

S"mmary

Introd"ction %verage elocity and .emerat"re

+aminar and ."rb"lent Flow in ."bes

.$e Dntrance


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Lect - 11 Internal Forced convection.pptx