lect - 11 internal forced convection.pptx
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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$e
t"be-
("sselt n"mber Dstimations
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5/12/16 | Slide 2C
."rb"lent Flow in ."bes
w$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$e second
(etukhov euation e'ressed as
("sselt n"mber Dstimations
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5/12/16 | Slide *>
."rb"lent Flow in ."bes
.$e acc"racy of t$is relation at lower
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5/12/16 | Slide *1
."rb"lent Flow in ."bes
For li,"id metals 4>->> = Hr = >->1! t$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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#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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