7/24/2019 Mec 551 Convection
1/174
1
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
7/24/2019 Mec 551 Convection
2/174
2
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.
7/24/2019 Mec 551 Convection
3/174
3
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..
7/24/2019 Mec 551 Convection
4/174
3.13.1 Convective !rinciplesConvective !rinciples
7/24/2019 Mec 551 Convection
5/174
"
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).
7/24/2019 Mec 551 Convection
6/174
*
Convection !rinciplesConvection !rinciples
#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/
7/24/2019 Mec 551 Convection
7/174
0
Convection !rinciplesConvection !rinciples
#he difference bet$een e'ternal and internal flo$s is
sho$n in the figure belo$%
+'ternal Flo$+'ternal Flo$
nternalnternal
Flo$Flo$
7/24/2019 Mec 551 Convection
8/174
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
7/24/2019 Mec 551 Convection
9/174
4
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, $hich group together indimensionless numbers 9 to reducedimensionless numbers 9 to reduce
the number of variables.the number of variables.
7/24/2019 Mec 551 Convection
10/174
1:
Convection !rinciplesConvection !rinciples(&usselt &umber)(&usselt &umber)
#he &usselt number is a#he &usselt number is a non6dimensionali8ed hnon6dimensionali8ed h,,
defined as%defined as%
k
hLNu c= //cc 6 Characteristic /ength6 Characteristic /ength
; 6 #hermal conductivity of fluid; 6 #hermal conductivity of fluid
7/24/2019 Mec 551 Convection
11/174
11
Convection !rinciplesConvection !rinciples(&usselt &umber)(&usselt &umber)
7/24/2019 Mec 551 Convection
12/174
12
Convection !rinciplesConvection !rinciples(&usselt &umber)(&usselt &umber)
#a;ing the ratio of these t$o equations%#a;ing the ratio of these t$o equations%
#hus &u represents the enhancement of heat transfer through a#hus &u represents the enhancement of heat transfer through a
fluid layer as a result of convection relative to conductionfluid layer as a result of convection relative to conduction
across the same fluid layer. #he larger &u, the more effectiveacross the same fluid layer. #he larger &u, the more effective
the convection.the convection.
&u> 1 for a fluid layer, represents pure conduction.&u> 1 for a fluid layer, represents pure conduction.
Nu
k
LhTh
q
q
LTkcond
conv =
=
=
7/24/2019 Mec 551 Convection
13/174
13
Convection !rinciplesConvection !rinciples(?iscosity)(?iscosity)
?iscosity?iscosity
A measure of the internal stic;iness ofA measure of the internal stic;iness of
the fluid. #he friction force bet$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,
7/24/2019 Mec 551 Convection
14/174
1
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$ay from the fluid.
@inematic viscosity@inematic viscosity(() 9) 9 #he ratio of dynamic viscosity#he ratio of dynamic viscosityto density.to density.
Convection !rinciplesConvection !rinciples(?iscosity)(?iscosity)
=
dydu
=
7/24/2019 Mec 551 Convection
15/174
1"
Convection !rinciplesConvection !rinciples(?iscosity)(?iscosity)
?iscous flo$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$s.
7/24/2019 Mec 551 Convection
16/174
1*
Convection !rinciplesConvection !rinciples(Compressibility)(Compressibility)
Compressible flo$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. $ater).
7as7as
/iquid/iquid
7/24/2019 Mec 551 Convection
17/174
10
Convection !rinciplesConvection !rinciples(#ypes of Flo$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.
7/24/2019 Mec 551 Convection
18/174
1
Convection !rinciplesConvection !rinciples(#ypes of Flo$s)(#ypes of Flo$s)
7/24/2019 Mec 551 Convection
19/174
14
Convection !rinciplesConvection !rinciples(#ypes of Flo$s)(#ypes of Flo$s)
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.
7/24/2019 Mec 551 Convection
20/174
2:
3.23.2 Convection boundaryConvection boundarylayer theorylayer theory
7/24/2019 Mec 551 Convection
21/174
21
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.
7/24/2019 Mec 551 Convection
22/174
22
Convection !rinciplesConvection !rinciples(?elocity oundary /ayer)(?elocity oundary /ayer)
?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 $here%
= uu 99.0
7/24/2019 Mec 551 Convection
23/174
23
Convection !rinciplesConvection !rinciples(?elocity oundary /ayer)(?elocity oundary /ayer)
#he dashed line, divides the flo$#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 $hich the friction effects are
negligible and the velocitynegligible and the velocity
remains constant.remains constant.
uEyy
''
Heated
7/24/2019 Mec 551 Convection
24/174
2
Convection !rinciplesConvection !rinciples(?elocity oundary /ayer)(?elocity oundary /ayer)
Flo$ regions in velocity boundary of a flat plate%Flo$ regions in velocity boundary of a flat plate%
7/24/2019 Mec 551 Convection
25/174
2"
Convection !rinciplesConvection !rinciples(?elocity oundary /ayer)(?elocity oundary /ayer)
Comparison of a laminar and turbulent velocityComparison of a laminar and turbulent velocity
boundary layer profile%boundary layer profile%
7/24/2019 Mec 551 Convection
26/174
2*
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
7/24/2019 Mec 551 Convection
27/174
20
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)
7/24/2019 Mec 551 Convection
28/174
2
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
7/24/2019 Mec 551 Convection
29/174
24
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).
7/24/2019 Mec 551 Convection
30/174
3:
Convection !rinciplesConvection !rinciples(eynolds &umber)(eynolds &umber)
#he eynolds number is defined as%#he eynolds number is defined as%
LVLV
ForcesViscousForcesnertia ===Re
? upstream velocity/ characteristic length
n > mrkinematic viscosity of fluid
7/24/2019 Mec 551 Convection
31/174
31
Convection !rinciplesConvection !rinciples(eynolds &umber)(eynolds &umber)
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
7/24/2019 Mec 551 Convection
32/174
32
Convection !rinciplesConvection !rinciples(eynolds &umber)(eynolds &umber)
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
7/24/2019 Mec 551 Convection
33/174
33
3.33.3 Forced convection over anForced convection over an
e'terior surfacee'terior surface(laminar and turbulent flo$)(laminar and turbulent flo$)
7/24/2019 Mec 551 Convection
34/174
3
+'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.
7/24/2019 Mec 551 Convection
35/174
3"
+'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
=
=
7/24/2019 Mec 551 Convection
36/174
3*
+'ternal Flo$ +quations+'ternal Flo$ +quations(Conservation of Iass)(Conservation of Iass)
ate of massate of mass
flo$ intoflo$ into
control volumecontrol volume
ate of massate of mass
flo$ intoflo$ into
control volumecontrol volume
ate of massate of mass
flo$ out offlo$ out of
control volumecontrol volume
ate of massate of mass
flo$ out offlo$ out of
control volumecontrol volume
>>
dyd!yvd!vdyd!
!udyud!vdyu
d!dyy
vvdyd!
!
uud!vdyu
++
+=+
++
+=+
0=
+
y
v
!
uJ 26 Continuity +quationJ 26 Continuity +quation
7/24/2019 Mec 551 Convection
37/174
30
+'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
7/24/2019 Mec 551 Convection
38/174
3
+'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
7/24/2019 Mec 551 Convection
39/174
34
+'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 &&
7/24/2019 Mec 551 Convection
40/174
:
+'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
7/24/2019 Mec 551 Convection
41/174
1
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$ ($ithreduce for steady, incompressible laminar flo$ ($ith
constant properties) over a flat plate.constant properties) over a flat plate.
'
y
##
, u, u
u(',:)> :u(',:)> :
v(',:)> :v(',:)> :
#(',:)> ##(',:)> #ss
dydy
d'd'
oundary layeroundary layer
7/24/2019 Mec 551 Convection
42/174
2
Convection over a Flat !lateConvection over a Flat !late
Consider elemental control volume for force balanceConsider elemental control volume for force balance
in the laminar boundary layer.in the laminar boundary layer.
Continuity%Continuity%
Iomentum%Iomentum%
+nergy%+nergy%
0
=
+
y
v
!
u
2
2
y
u
y
uv
!
uu
=
+
2
2
y
T
y
Tv
!
Tu
=
+
7/24/2019 Mec 551 Convection
43/174
7/24/2019 Mec 551 Convection
44/174
Convection over a Flat !lateConvection over a Flat !late
ecall, that the stream function is defined as%ecall, that the stream function is defined as%
ependent variable%ependent variable%
!v
yu
=
=
;
( )yu
u!u
f
=
=
7/24/2019 Mec 551 Convection
45/174
"
Convection over a Flat !lateConvection over a Flat !late
#herefore%#herefore%
=
=
=
=
=
=
=
=
fd
df
!
u
f!u
u
d!
df
u
!u
!!v
d
dfu
!
u
d
df
u
!u
yyu
2
1
2
7/24/2019 Mec 551 Convection
46/174
*
Convection over a Flat !lateConvection over a Flat !late
7/24/2019 Mec 551 Convection
47/174
0
Convection over a Flat !lateConvection over a Flat !late(Iomentum +quation)(Iomentum +quation)
7/24/2019 Mec 551 Convection
48/174
Convection over a Flat !lateConvection over a Flat !late(Iomentum +quation)(Iomentum +quation)
sing the definitions for f andsing the definitions for f and M, the boundary equations inM, the boundary equations in
terms of the similarity variables can be found.terms of the similarity variables can be found.
( )
1
0
00
0
=
=
=
=
=
d
df
d
df
f Ho$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
7/24/2019 Mec 551 Convection
49/174
4
Convection over a Flat !lateConvection over a Flat !late(Iomentum +quation)(Iomentum +quation)
#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 $hen%
992.0==u
u
d
df
0.5=
99.0=uu
C ti Fl t !l tC ti Fl t !l t
7/24/2019 Mec 551 Convection
50/174
":
Convection over a Flat !lateConvection over a Flat !late(Iomentum +quation)(Iomentum +quation)
7/24/2019 Mec 551 Convection
51/174
"1
+nergy +quation+nergy +quation
@no$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!
=
,,
C ti Fl t !l tC ti Fl t !l t
7/24/2019 Mec 551 Convection
52/174
"2
Convection over a Flat !lateConvection over a Flat !late(+nergy +quation)(+nergy +quation)
N(M)
7/24/2019 Mec 551 Convection
53/174
"3
Convection over a Flat !lateConvection over a Flat !late(+nergy +quation)(+nergy +quation)
7/24/2019 Mec 551 Convection
54/174
"
Convection over a Flat !lateConvection over a Flat !late(+nergy +quation)(+nergy +quation)
and%and% ( )
yud
df
yu
u
!u
f
=
=
=
Con ection o er a Flat !lateConvection over a Flat !late
7/24/2019 Mec 551 Convection
55/174
""
Convection over a Flat !lateConvection over a Flat !late(+nergy +quation)(+nergy +quation)
7/24/2019 Mec 551 Convection
56/174
"*
Convection over a Flat !lateConvection over a Flat !late(+nergy +quation)(+nergy +quation)
=
+
!
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 =
+
Convection over a Flat !lateConvection over a Flat !late
7/24/2019 Mec 551 Convection
57/174
"0
Convection over a Flat !lateConvection over a Flat !late(+nergy +quation)(+nergy +quation)
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
Convection over a Flat !lateConvection over a Flat !late
7/24/2019 Mec 551 Convection
58/174
"
Convection over a Flat !lateConvection over a Flat !late(+nergy +quation)(+nergy +quation)
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
Convection over a Flat !lateConvection over a Flat !late
7/24/2019 Mec 551 Convection
59/174
"4
Convection over a Flat !lateConvection over a Flat !late(+nergy +quation)(+nergy +quation)
#he temperature gradient at the surface is%#he temperature gradient at the surface is%
7/24/2019 Mec 551 Convection
60/174
*:
Convection over a Flat !lateConvection over a Flat !late(+nergy +quation)(+nergy +quation)
#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
Convection over a Flat !lateConvection over a Flat !late
7/24/2019 Mec 551 Convection
61/174
*1
#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
Convection over a Flat !lateConvection over a Flat !late
7/24/2019 Mec 551 Convection
62/174
*2
Convection over a Flat !lateConvection over a Flat !late(/aminar Flo$)(/aminar Flo$)
#he local friction coefficient (C#he local friction coefficient (CF'F') can also be determined.) can also be determined.
7/24/2019 Mec 551 Convection
63/174
*3
Convection over a Flat !lateConvection over a Flat !late(/aminar Flo$)(/aminar Flo$)
#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
Convection over a Flat !lateConvection over a Flat !late
7/24/2019 Mec 551 Convection
64/174
*
Convection over a Flat !lateConvection over a Flat !late(/aminar Flo$)(/aminar Flo$)
#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
=
=
=
=
Convection over a Flat !lateConvection over a Flat !late
7/24/2019 Mec 551 Convection
65/174
*"
Convection over a Flat !lateConvection over a Flat !late(/aminar Flo$)(/aminar Flo$)
7/24/2019 Mec 551 Convection
66/174
Convection over a Flat !lateConvection over a Flat !late
7/24/2019 Mec 551 Convection
67/174
*0
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$s over it at ms. #he plate is ;ept at a surface
temperature (#temperature (#ss> : QC).> : QC).
": m": m
''
y
: QC
ms
#s> :QC
Convection over a Flat !lateConvection over a Flat !late
7/24/2019 Mec 551 Convection
68/174
*
Convection over a Flat !lateConvection over a Flat !late(+'ample 3.1)(+'ample 3.1)
#he first step is to calculate the mean film temperature of the#he first step is to calculate the mean film temperature of the
fluid flo$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
Convection over a Flat !lateConvection over a Flat !late
7/24/2019 Mec 551 Convection
69/174
*4
Convection over a Flat !lateConvection over a Flat !late(+'ample 3.1)(+'ample 3.1)
For liquid $ater at *:For liquid $ater at *: QC from #able A64 in the te't boo;%QC from #able A64 in the te't boo;%
": m": m''
yy
: QC
: QC
99.2Pr
654.0
67.4
3.983 3
=
=
=
=
Cm(
sm
k%
m
k%
k
7/24/2019 Mec 551 Convection
70/174
Convection over a Flat !lateConvection over a Flat !late
7/24/2019 Mec 551 Convection
71/174
01
Convection over a Flat !lateConvection over a Flat !late(+'ample 3.1)(+'ample 3.1)
+'ample 3.1b+'ample 3.1b &o$ 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.
7/24/2019 Mec 551 Convection
72/174
02
Convection over a Flat !lateConvection over a Flat !late(+'ample 3.1)(+'ample 3.1)
#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
Convection over a Flat !lateConvection over a Flat !late
7/24/2019 Mec 551 Convection
73/174
03
Convection over a Flat !lateConvection over a Flat !late(+'ample 3.1)(+'ample 3.1)
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
==
=
Convection over a Flat !lateConvection over a Flat !late
7/24/2019 Mec 551 Convection
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$
Convection over a Flat !lateConvection over a Flat !late
7/24/2019 Mec 551 Convection
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
Convection over a Flat !lateConvection over a Flat !late
7/24/2019 Mec 551 Convection
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.
Convection over a Flat !lateConvection over a Flat !late(+'ample 3.2)(+'ample 3.2)
: m
uu
> " ms> " ms
##> 2"> 2"QCQC
##ss> "> "QCQC
Convection over a Flat !lateConvection over a Flat !late
7/24/2019 Mec 551 Convection
77/174
00
Convection over a Flat !lateConvection over a Flat !late(+'ample 3.2)(+'ample 3.2)
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
=
==
=
7/24/2019 Mec 551 Convection
78/174
Convection over a Flat !lateConvection over a Flat !late
7/24/2019 Mec 551 Convection
79/174
04
Convection over a Flat !lateConvection over a Flat !late(+'ample 3.2)(+'ample 3.2)
#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
==
=
7/24/2019 Mec 551 Convection
80/174
Convection over a Flat !lateConvection over a Flat !late
7/24/2019 Mec 551 Convection
81/174
1
Convection over a Flat !lateConvection over a Flat !late(+'ample 3.2)(+'ample 3.2)
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
Convection over a Flat !lateConvection over a Flat !late
7/24/2019 Mec 551 Convection
82/174
2
Convection over a Flat !lateConvection over a Flat !late(+'ample 3.2)(+'ample 3.2)
#he total heat flu' per is%#he total heat flu' per is%
( )
( ) ( ) ( )(
CCmm
TTAhQ
Cm
(
ss
960,54
25451407.68 2
=
=
=
7/24/2019 Mec 551 Convection
83/174
Forced ConvectionForced Convection
7/24/2019 Mec 551 Convection
84/174
Forced Convectiono ced Co ect o(on Cylinders and
7/24/2019 Mec 551 Convection
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)%
Forced ConvectionForced Convection
7/24/2019 Mec 551 Convection
86/174
*
Forced Convection(over Circular and &on6Circular Cylinders)(over Circular and &on6Circular Cylinders)
Forced ConvectionForced Convection
7/24/2019 Mec 551 Convection
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
Forced ConvectionForced Convection
7/24/2019 Mec 551 Convection
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
=
==
Forced ConvectionForced Convection
7/24/2019 Mec 551 Convection
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
Forced ConvectionForced Convection
7/24/2019 Mec 551 Convection
90/174
4:
(+'ample 3.3)(+'ample 3.3)
#herefore%#herefore%
#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
7/24/2019 Mec 551 Convection
91/174
+'ample 1+'ample 1
7/24/2019 Mec 551 Convection
92/174
+'ample 1+'ample 1
42
+'ample 2+'ample 2
7/24/2019 Mec 551 Convection
93/174
+'ample 2+'ample 2
43
+'ample 3+'ample 3
7/24/2019 Mec 551 Convection
94/174
+'ample 3+'ample 3
4
7/24/2019 Mec 551 Convection
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
7/24/2019 Mec 551 Convection
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
7/24/2019 Mec 551 Convection
97/174
40
convection equationsconvection equations
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
7/24/2019 Mec 551 Convection
98/174
4
convection equationsconvection equations
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
, #, #
7/24/2019 Mec 551 Convection
99/174
44
7/24/2019 Mec 551 Convection
100/174
1::
7/24/2019 Mec 551 Convection
101/174
1:1
7/24/2019 Mec 551 Convection
102/174
1:2
7/24/2019 Mec 551 Convection
103/174
1:3
7/24/2019 Mec 551 Convection
104/174
7/24/2019 Mec 551 Convection
105/174
1:"
3."3." eynolds Analogyeynolds Analogy
Forced ConvectionForced Convection
7/24/2019 Mec 551 Convection
106/174
1:*
(rag Force)(rag Force)
Forced ConvectionForced Convection( ld A l )
7/24/2019 Mec 551 Convection
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 )
7/24/2019 Mec 551 Convection
108/174
1:
(eynolds Analogy)(eynolds Analogy)
7/24/2019 Mec 551 Convection
109/174
1:4
(eynolds Analogy)(eynolds Analogy)
( )
( )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
=
=
=
===
=
7/24/2019 Mec 551 Convection
110/174
11:
(eynolds Analogy)(eynolds Analogy)
7/24/2019 Mec 551 Convection
111/174
111
(eynolds Analogy)(eynolds Analogy)
7/24/2019 Mec 551 Convection
112/174
112
(eynolds Analogy)(eynolds Analogy)
#herefore%#herefore%
( )
( )
0*
*
*
0*
*
*
=
=
=
=
y
ys
s
y
T
L
ky
T
TT!
TTkh
Forced ConvectionForced Convection( ld A l )( ld A l )
7/24/2019 Mec 551 Convection
113/174
113
7/24/2019 Mec 551 Convection
114/174
11
(eynolds Analogy)(eynolds Analogy)
&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)
7/24/2019 Mec 551 Convection
115/174
11"
(eynolds Analogy)(eynolds Analogy)
#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 $ith a minimum amount of e'periments and
report their results in terms of Dust e and !r.report their results in terms of Dust e and !r.
( ) ( )Pr,ReRe 34 LLF %NuandfC ==
Forced ConvectionForced Convection( ld A l )(eynolds Analogy)
7/24/2019 Mec 551 Convection
116/174
11*
(eynolds Analogy)(eynolds Analogy)
#he e'perimental data for heat transfer is often#he e'perimental data for heat transfer is oftenrepresented ($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)
7/24/2019 Mec 551 Convection
117/174
110
(eynolds Analogy)(eynolds Analogy)
7/24/2019 Mec 551 Convection
118/174
11
(eynolds Analogy)(eynolds Analogy)
&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 $hen u > u
> ?> ?
> constant)> constant)
0*
*
=!
#
For !r > 1, theFor !r > 1, the
thermal andthermal andvelocity boundaryvelocity boundary
layers coincidelayers coincide
Forced ConvectionForced Convection(eynolds Analogy)(eynolds Analogy)
7/24/2019 Mec 551 Convection
119/174
114
(eynolds Analogy)(eynold s Analogy)
#he equations then become%#he equations then become%
Momentum:Momentum:
Energy:Energy:
&ote% #hese t$o equations are e'actly in the same&ote% #hese t$o equations are e'actly in the same
form for uV and #V.form for uV and #V.
2*
*2
*
**
*
**
2*
*2
*
**
*
**
Re
1
Re
1
y
T
y
Tv
!
Tu
y
u
y
uv
!
uu
L
L
=
+
=
+
Forced ConvectionForced Convection(eynolds Analogy)(eynolds Analogy)
7/24/2019 Mec 551 Convection
120/174
12:
7/24/2019 Mec 551 Convection
121/174
121
(eynold s Analogy)(eynold s Analogy)
7/24/2019 Mec 551 Convection
122/174
122
(eynold s Analogy)(eynold s Analogy)
#herefore substituting these values into#herefore substituting these values into +quation+quationVVgives%gives%
0*
*
*
0*
*
*
==
=
yy y
T
y
u
2
2
Re
,
,
!F
!
!F
!
C)t
or
CNu
=
= eynolds Analogy for
!r > 1
Forced ConvectionForced Convection(
7/24/2019 Mec 551 Convection
123/174
123
(
7/24/2019 Mec 551 Convection
124/174
12
(eynold s Analogy)(eynold s Analogy)
eynolds Analogy is important because it allo$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)
7/24/2019 Mec 551 Convection
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 $ide range of !r.
#his is done by adding a#his is done by adding a !randtl number correction!randtl number correction..
0*
*
=!
#
Forced ConvectionForced Convection(Chilton6Colburn Analogy)(Chilton6Colburn Analogy)
7/24/2019 Mec 551 Convection
126/174
12*
(Chilton6Colburn Analogy)(Chilton6Colburn Analogy)
ecall as previously derived%ecall as previously derived%
#a;ing their ratio and rearranging give the relation#a;ing their ratio and rearranging give the relation
;no$n as the;no$n as the Chilton6Colburn analogyChilton6Colburn analogyor theor the
modified eynoldWs analogymodified eynoldWs analogy%%
21
31
21
RePr332.0Re664.0, !!!!F NuandC ==
HL!
!F*Nu
C== 1, RePr
2
31
32
Pr2
,
==VC
hC*
p
!!F
H
For :.* R !r R *:
Colburn D6factorColburn D6factor
Forced ConvectionForced Convection(Chilton6Colburn Analogy)(Chilton6Colburn Analogy)
7/24/2019 Mec 551 Convection
127/174
120
(Chilton6Colburn Analogy)(Chilton6Colburn Analogy)
#he Chilton6Colburn Analogy is derived using%#he Chilton6Colburn Analogy is derived using% /aminar flo$/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 )
7/24/2019 Mec 551 Convection
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 )
7/24/2019 Mec 551 Convection
129/174
124
(+'ample 3.)(+'ample 3.)
3 m J Characteristic length 3 m J Characteristic length
7/24/2019 Mec 551 Convection
130/174
13:
For all flat plates%For all flat plates%
Drag = Friction ForceDrag = Friction Force
#herefore%#herefore%
(+'ample 3.)(+'ample 3.)
( )
( ) ( ) ( ) 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.)
7/24/2019 Mec 551 Convection
131/174
131
(+'ample 3.)(+'ample 3.)
#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
7/24/2019 Mec 551 Convection
132/174
132
3.*3.* Convection in anConvection in an
internal flo$internal flo$
7/24/2019 Mec 551 Convection
133/174
nternal Flo$nternal Flo$(&on6Circular #ubes)(&on6Circular #ubes)
7/24/2019 Mec 551 Convection
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)
7/24/2019 Mec 551 Convection
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)
7/24/2019 Mec 551 Convection
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)
7/24/2019 Mec 551 Convection
137/174
130
( g )( g )
nternal Flo$ +quationsnternal Flo$ +quations(+'ample 3.")(+'ample 3.")
7/24/2019 Mec 551 Convection
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
nternal Flo$ +quationsnternal Flo$ +quations(+'ample 3.")(+'ample 3.")
7/24/2019 Mec 551 Convection
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
nternal Flo$ +quationsnternal Flo$ +quations(+'ample 3.")(+'ample 3.")
7/24/2019 Mec 551 Convection
140/174
1:
( p )( p )
(a) Find the temperature and velocity distributions :
Continuity +quation%Continuity +quation%
#he '6component of velocity does not change.
7/24/2019 Mec 551 Convection
141/174
11
( p )( p )
'6momentum equation%'6momentum equation%
#his is a 2#his is a 2ndndorder differential equation.
7/24/2019 Mec 551 Convection
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
nternal Flo$ +quationsnternal Flo$ +quations(+'ample 3.")(+'ample 3.")
7/24/2019 Mec 551 Convection
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 $ith y only #> #(y).
VL
yu =
nternal Flo$ +quationsnternal Flo$ +quations(+'ample 3.")(+'ample 3.")
7/24/2019 Mec 551 Convection
144/174
1
:: ::
:: :: ::
( p )( p )
7/24/2019 Mec 551 Convection
145/174
1"
( p )
2
2
2
=
LV
yTk
7/24/2019 Mec 551 Convection
146/174
1*
( )
&o$ integrating the equation t$ice%&o$ integrating the equation t$ice%
Applying boundary conditions%Applying boundary conditions%
#(:) > ##(:) > #::
#(/) > ##(/) > #::
43
2
2CyCV
L
y
kT ++
=
40:0 CTy ==
2
3
03
2
0
2
2:
VkL
C
TLCVL
L
kTLy
=
++
==
7/24/2019 Mec 551 Convection
147/174
nternal Flo$ +quationsnternal Flo$ +quations(+'ample 3.")(+'ample 3.")
7/24/2019 Mec 551 Convection
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
====
7/24/2019 Mec 551 Convection
149/174
nternal Flo$ +quationsnternal Flo$ +quations(+'ample 3.")(+'ample 3.")
7/24/2019 Mec 551 Convection
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
=
=
=
==
=
nternal Flo$ +quationsnternal Flo$ +quations(+'ample 3.")(+'ample 3.")
7/24/2019 Mec 551 Convection
151/174
1"1
As a chec;, $e can also calculate the heat flu' at y> / (shouldAs a chec;, $e can also calculate the heat flu' at y> / (shouldbe equal but opposite sign).be equal but opposite sign).
( ) ( )( )
( )
( )
2
22
2
800,28
1
1
002.02
128.0
2
21
2
2
m
(
(
mL
V
L
L
kL
Vk
dy
dTkq
smN
sm
m
sN
Ly
L
+=
+=
+=
==
=
Correct Y
nternal Flo$ +quationsnternal Flo$ +quations(+'ample 3.")(+'ample 3.")
7/24/2019 Mec 551 Convection
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
7/24/2019 Mec 551 Convection
153/174
7/24/2019 Mec 551 Convection
154/174
1"
3.03.0 Free (natural) convectionFree (natural) convection
7/24/2019 Mec 551 Convection
155/174
Free ConvectionFree Convection(?olume +'pansion Coefficient)(?olume +'pansion Coefficient)
7/24/2019 Mec 551 Convection
156/174
1"*
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) $hich is
defined as%defined as%
## TT
=
=
11
Free ConvectionFree Convection(?olume +'pansion Coefficient)(?olume +'pansion Coefficient)
7/24/2019 Mec 551 Convection
157/174
1"0
n natural convection studies, the condition of the fluidn natural convection studies, the condition of the fluidsufficiently far from the hot or cold surface is indicated by thesufficiently far from the hot or cold surface is indicated by the
subscript [subscript [
\ to indicate that the presence of the surface is not\ to indicate that the presence of the surface is not
felt.felt.
n such cases,n such cases, Z can be e'pressed appro'imately by replacingZ can be e'pressed appro'imately by replacing
the differential equations by differences, such as%the differential equations by differences, such as%
( )
( )TTT =
=
11
( ) = TT
Free ConvectionFree Convection(?olume +'pansion Coefficient)(?olume +'pansion Coefficient)
7/24/2019 Mec 551 Convection
158/174
1"
For an ideal gas%For an ideal gas%
#hus for an#hus for an ideal gasideal gasthe discharge coefficientthe discharge coefficientbecomes%becomes%
T,
#
=
TTT,T#
,T#
#
111
=
=
=
Free ConvectionFree Convection(7rashof &umber)(7rashof &umber)
7/24/2019 Mec 551 Convection
159/174
1"4
#he velocity and temperature for natural#he velocity and temperature for naturalconvectionconvection over a vertical plateover a vertical plateareare
sho$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 $ell as the
surface of the plate.surface of the plate.
#his is e'pected since the fluid#his is e'pected since the fluid
beyond the boundary layer isbeyond the boundary layer is
motionless.motionless.
Free ConvectionFree Convection(7rashof &umber)(7rashof &umber)
7/24/2019 Mec 551 Convection
160/174
1*:
ecall that the '6momentum equations is%ecall that the '6momentum equations is%
&o$ 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
%!
#=
Free ConvectionFree Convection(7rashof &umber)(7rashof &umber)
7/24/2019 Mec 551 Convection
161/174
1*1
7/24/2019 Mec 551 Convection
162/174
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
Free ConvectionFree Convection(7rashof &umber)(7rashof &umber)
7/24/2019 Mec 551 Convection
163/174
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 =
Free ConvectionFree Convection(7rashof &umber)(7rashof &umber)
7/24/2019 Mec 551 Convection
164/174
1*
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)
7/24/2019 Mec 551 Convection
165/174
1*"
/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.*)
7/24/2019 Mec 551 Convection
166/174
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
Free ConvectionFree Convection(+'ample 3.*)(+'ample 3.*)
7/24/2019 Mec 551 Convection
167/174
1*0
Assume%Assume%
7/24/2019 Mec 551 Convection
168/174
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==
Free ConvectionFree Convection(+'ample 3.*)(+'ample 3.*)
7/24/2019 Mec 551 Convection
169/174
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
Free ConvectionFree Convection(+'ample 3.*)(+'ample 3.*)
7/24/2019 Mec 551 Convection
170/174
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%
Free ConvectionFree Convection(+'ample 3.*)(+'ample 3.*)
7/24/2019 Mec 551 Convection
171/174
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
Free ConvectionFree Convection(+'ample 3.*)(+'ample 3.*)
7/24/2019 Mec 551 Convection
172/174
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
== =
Free ConvectionFree Convection(+'ample 3.*)(+'ample 3.*)
7/24/2019 Mec 551 Convection
173/174
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
7/24/2019 Mec 551 Convection
174/174
Top Related