Laminar Flow Friction and Heat Transfer in Non- Circular Ducts and Channels Part II- Thermal
Problem
Y.S. Muzychka and M.M. Yovanovich
Compact Heat Exchangers
A Festschrift on the 60 th Birthday of Ramesh K. Shah
Grenoble, France August 24, 2002
Proceedings Editors
G.P. Celata, B. Thonon, A. Bontemps, S. Kandlikar
Pages 131-139
IN NON-CIRCULAR DUCTS AND CHANNELS
PART II - THERMAL PROBLEM
Y.S.Muzychka
and M.M.Yovanovich Æ
FacultyofEngineeringandAppliedScience,MemorialUniversityofNewfoundland,St. John's,
NF,Canada,A1B3X5, Æ
DepartmentofMechanicalEngineering,UniversityofWaterloo,
Waterloo,ON,Canada,N2L3G1
ABSTRACT
Adetailedreviewandanalysisofthethermalcharacteristicsoflaminardevelopingandfullydevelopedowinnon-circular
ducts is presented. New models are proposed which simplifythe prediction of Nusselt numbersfor threefundamental
ows: thecombinedentranceproblem,theGraetzproblem,andthermallyfullydevelopedowinmostnon-circularduct
geometriesfoundinheatexchangerapplications. Bymeansofscalinganalysisitisshownthatthecompleteproblemmay
beeasilyanalyzedbycombiningtheasymptoticresultsforshortandlongducts. Bymeansofanewcharacteristiclength
scale,thesquarerootofcross-sectionalarea,theeectofductshapehasbeenreduced. Thenewmodelhasanaccuracy
of10percent,orbetter,formostcommonductshapes. Bothsinglyanddoublyconnectedductsareconsidered.
NOMENCLATURE
A = owarea,m
2
a;b = majorandminoraxesofellipseorrectangle,m
B
i
= constants,i=1:::4
C = constant
C
i
= constants,i=1:::5
c
p
= specicheat,J=kgK
D
h
= hydraulicdiameterofplainchannel,4A=P
f = frictionfactor=(
1
2 U
2
)
h = heattransfercoeÆcient,W=m 2
K
k = thermalconductivity, W=mK
L = ductlength,m
L
= dimensionlessthermallength,L=LR e
L Pr
L = characteristiclengthscale,m
m = correlationparameter
_
m = massowrate,kg=s
n = inwarddirectednormal
Nu
L
= Nusseltnumber,hL=k
P = perimeter,m
Pr = Prandtlnumber,=
q = heatux,W=m
2
r = radius,m
R e
L
= Reynoldsnumber,UL=
s = arclength,m
T = temperature,K
u;v;w = velocitycomponents,m=s
U = averagevelocity,m=s
x;y;z = cartesiancoordinates,m
Y = dimensionlesscoordinate,y=L
z = axialcoordinate,m
Z = dimensionlesspositionforthermally
developingows,z=LR e
L Pr
Greek Symbols
= thermaldiusivity,m 2
=s
Æ = hydrodynamicboundarylayerthickness,m
= thermalboundarylayerthickness,m
= aspectratio,b=a
= symmetryparameter
= dynamicviscosity,Ns=m 2
= kinematicviscosity,m 2
=s
= uiddensity,kg=m 3
= wallshearstress,N=m 2
= temperatureexcess,T T
o ,K
Subscripts
p
A = basedonthesquareroot ofarea
D
h
= basedonthehydraulicdiameter
H = basedonisouxcondition
h = hydrodynamic
L = basedonthearbitrarylengthL
m = mixedorbulkvalue
T = basedonisothermalcondition
t = thermal
Superscripts
() = denotesaveragevalueof()
INTRODUCTION
Heat transfer in non-circular ducts of constant cross-
sectionalareaisofparticularinterestinthedesignofcom-
pact heat exchangers. In these applications passages are
generallyshortandusuallycomposedofcross-sectionssuch
as triangularor rectangular geometries in addition to the
circular tube or parallel plate channel. Also, due to the
wide range of applications, uid Prandtl numbers usually
vary between0:1<Pr<1,which coversawiderangeof
uids encompassing gasesand highly viscous liquids such
as automotiveoils. In the second part of this paper, the
authors developmodelsforthermallyfullydevelopedow,
thermally developing or Graetz ow, and simultaneously
developingowincircularandnon-circularductsandchan-
nels.
A review of the literature [1,2] reveals that the only
models available for predicting heat transfer in the com-
forthecircularductandStephan[5,6]fortheparallelplate
channelandcircularduct. Recently,Garimellaetal. [7]de-
veloped empiricalexpressions for therectangular channel,
whilenumericaldataforpolygonalductswereobtainedby
Asakoet al. [8]. Additional data forthe rectangular, cir-
cular, triangular, and parallel plate channel are available
in Shah and London [1], Rohsenow et al. [9], and Kakac
and Yener[10]. Yilmaz and Cihan [11,12]provide acom-
plexcorrelationmethodforthecaseofthermallydeveloping
owwithfullydevelopedhydrodynamicow,oftenreferred
toastheGraetzproblemorGraetzow. Morerecently,the
authorshavepresentedsimplermodelsforboththeGraetz
problem[13],thecombinedentranceproblem[14],andnat-
uralconvectioninverticalisothermalducts[15].Additional
resultsarereportedinMuzychka[16].
InPartIofthispaper,thehydrodynamicproblemwas
consideredin detail. PartIIofthispaperwill presentnew
models for the associated thermal problem. These mod-
elshavebeendevelopedusingtheChurchillandUsagi[17]
asymptoticcorrelationmethod,aswasdoneinPartIforthe
hydrodynamicproblem. In thispaper,the asymptoticso-
lutionsforthermallyfully developedow,L>>L
h
;L>>
L
t
, thermally developing ow, L >> L
h
;L << L
t , and
thecombinedentryproblem,L<<L
h
;L<<L
t
,areused
todevelopamoregeneralmodel forpredictingheattrans-
fer coeÆcientsin non-circular ducts. Here L
t
denotes the
thermal entry length and L
h
the hydrodynamic entrance
length.
GOVERNING EQUATIONS
Thegoverningequationsforsteady,constantproperty,
incompressible ow, in the thermal entrance region in a
non-circularductorchannelare:
r
~
V =0 (1)
~
V r
~
V = rp+r 2
~
V (2)
c
p
~
V rT =kr 2
T (3)
Simultaneous solutionof the continuity, Eq. (1), and
momentum,Eq. (2), equationssubjectto thenoslip con-
dictionattheductwall,
~
V =0,theboundednesscondition
along the duct duct axis,
~
V 6= 1, and a constant initial
velocity,
~
V = U
~
k, are required to characterize the ow.
Intheenergy equation,the owis subjectto constantin-
lettemperature,theboundednessconditionalongtheduct
axis,and either uniformwalltemperature (UWT)usually
denoted witha subscript(T) or uniform wallux (UWF)
condition,usuallydenotedwithasubscript(H),attheduct
wall.
Inthenextsection,scalinganalysis[18]isusedtoshow
theappropriateformofthesolutionforbothshortandlong
ducts. Later, asymptoticanalysis[19] isusedto developa
newmodel.
SCALEANALYSIS
Wenowexaminetheenergyequationandconsiderthe
various balances implied under particular ow conditions.
verseconductionandaxialconvection,i.e.,
c
p
~
V rT
| {z }
Convection
= kr 2
T
| {z }
Conduction
(4)
Wenowconsiderseveralbalancesbyexaminingthein-
teractionofthehydrodynamicandthermalboundarylayers
in tworegions: thelongductand theshort duct. Eachis
examined below using the method of scale analysis advo-
catedbyBejan[18].
Long Duct Asymptote,L>>L
t
;L>>L
h
We beginby consideringthermallyand hydrodynami-
callyfullydevelopedowinanon-circularductofconstant
cross-section. Wewritetheleft handsideofEq. (4)as:
c
p
~
V rT c
p U
(T
w T
o )
L
(5)
Next considering anenthalpybalance ontheduct, we
write:
qPL=mc_
p (T
w T
o
) (6)
Using theaboverelationshipinEq. (5)weobtainthe
followingresult:
c
p
~
V rT qP
A
(7)
The energyequation, Eq. (4) forfully developed ow
nowscalesaccordingto
qP
A k
(T
w T
o )
L 2
(8)
whereLrepresentsacharacteristictransversallengthscale
oftheductcross-section. Thisallowsthefollowingrelation
using thegeometry:
Nu
L
= qL
k(T
w T
o )
A
PL
(9)
IfthechracteristiclengthscaleischosentobeL= p
A,as
donein PartI ofthis paper,then thefollowingexpression
isobtained:
Nu p
A
=B p
A
P
=B
1
(10)
Inotherwords,theNusseltnumberforfully developed
ow, scales to a constant related to the geometry of the
duct cross-section. This result is valid for both parabolic
oruniformvelocitydistributions.
Short DuctAsymptote,L<<L
t
;L>>L
h
Intheentranceregionneartheductinlettwoseparate
problems must be considered. The rst assumes that a
fully developed hydrodynamicboundarylayerexists while
thesecondconsidersthemoregeneralproblemwhereboth
hydrodynamicandthermalboundarylayersdevelop.
Beginning with the classical Graetz ow problem,
where thevelocity distribution is assumed to be fully de-
velopedandofparabolicdistribution,wemaywritetheleft
hand sideofEq. (4)as:
c
p
~
V rT c
p U
L
w o
L
(11)
where is the thermal boundary layer thickness. This
assumesthat thethermal boundary layeris conned to a
regionneartheductwallwherethevelocitydistributionis
approximatelylinear,i.e. j
~
VjU=L. Theenergyequa-
tionforGraetz ownowscalesaccordingto
c
p U
L (T
w T
o )
L
k (T
w T
o )
2
(12)
where thebulktemperatureis takento beequaltheinlet
temperature, due to a thin thermal boundary layer. Re-
arrangingthisexpressionfor yields:
L
L
LR e
L Pr
1=3
(13)
Next,considering theheattransfercoeÆcientwhich scales
accordingto
h(T
w T
o )
k(T
w T
o )
(14)
weobtainthefollowingexpressionfortheNusseltnumber:
Nu
L
= hL
k
1
(L
) 1=3
(15)
or
Nu
L
= B
2
(L
) 1=3
(16)
whereL
=L=LR e
L
Pr, isthedimensionless thermalduct
length.
Alternatively,wecouldhaveused:
~
V r
~
V
~
fR e
L U
L
(17)
where we haveused the result from Part I, i.e. Eq. (9),
forthedimensionlessaveragewallshear. Thisleadsto the
followingwellknownbehaviour:
L
L=fR e
L
LR e
L Pr
1=3
(18)
and
Nu
L
=B
2
fR e
L
L
1=3
(19)
BothEq. (16) andEq. (19)havethesamecharacter-
isticandorder ofmagnitude,sincefR e
L
isconstant.
Short Duct Asymptote,L<<L
t
;L<<L
h
Finally,inthecombinedentranceregion,thehydrody-
namicboundarylayerthicknessscalesaccordingtothewell
knownexpression[18]:
Æ
L
1
(R e
L )
1=2
(20)
Wenowconsidertwodistinctregions. Theseare>>
Æ and Æ >> . When >> Æ,
~
V U, and the energy
equationscalesaccordingto
c
p U
w o
L
k
w o
2
(21)
whichgives
L
1
(R e
L Pr)
1=2
(22)
andusing Eq. (14),theNusseltnumberbecomes:
Nu
L
= B
3
(L
) 1=2
(23)
When << Æ,
~
V U=Æ, and the energy equation
scalesaccordingto
c
p U
Æ (T
w T
o )
L
k (T
w T
o )
2
(24)
whichgives
L
1
R e 1=2
L Pr
1=3
(25)
andusing Eq. (14),theNusseltnumberbecomes:
Nu
L
= B
4
Pr 1=6
(L
) 1=2
(26)
In summary, we havefound from scaling analysis the
followingrelationshipsforthelocaloraverageNusseltnum-
ber:
Nu
L
= 8
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
:
B
1
L>>L
t
;L
h
B
2
fR e
L
L
1=3
L<<L
t
;L>>L
h
B
3
(L
) 1=2
L<<L
t
;L
h
,>>Æ
B
4
Pr 1=6
(L
) 1=2
L<<L
t
;L
h
,<<Æ (27)
This asymptotic behaviour will be examined further
andwillformthebasisforthenewmodeldevelopedforthe
thermal entrance problem. Finally, anexpression relating
theapproximatemagnitudeofthethermalentrancelength
may be obtained by considering an equality between two
asymptotic limits given in Eq. (27) for Graetz ow with
L
=L
t :
B
1
=B
2
fR e
L
L
t
1=3
(28)
or
L
t
=
B
2
B
1
3
fR e
L
(29)
Later,itwillbeshownthatthisapproximatescalecom-
pares wellwithexactsolutions.
ASYMPTOTIC ANALYSIS
In this section, asymptotic analysis [19] is usedto es-
tablish expressions for the characteristic long duct and
ysis. First, the long duct limit is considered yielding a
simple relationship for predicting Nusselt number. Addi-
tionally, the issue of an appropriate characteristic length
scaleisaddressed. Finally,theshortductlimitsareconsid-
eredbyre-examining theapproximate solutionmethod of
Leveque[20] andthelaminarboundarylayersolutionsfor
isothermalandisouxplates[1,2,18,19].
Long DuctAsymptote,L>>L
t
;L>>L
h
Thefullydevelopedowlimitfor bothhydrodynamic
andthermalproblemshasbeenaddressedbyMuzychka[16]
and reported in Muzychka andYovanovich [13,14]. Addi-
tionalresultsappearinYovanovichetal. [15]. Theserefer-
ences[13-15]providemodelsfortheclassicGraetzproblem
and the combined entrance problem for forced ow, and
fornaturalconvectionin vertical isothermalducts ofnon-
circularcross-section.
In Part I of this paper the authors have shown that
whenthefrictionfactor-Reynoldsnumberproductisbased
onthesquarerootofcross-sectionalarea,thelargenumber
ofdatawerereducedto asinglecurvewhich wasasimple
function of the aspect ratio of the duct or channel. We
alsoshowedthat this curvewasaccurately representedby
thersttermoftheseriessolutionfortherectangularduct
cross-section. Thisimportantresultisgivenby
fR e p
A
=
12
p
(1+)
1 192
5
tanh
2
(30)
Next, Muzychka [16] and Muzychka and Yovanovich
[13]appliedthesamereasoningtothefullydevelopedNus-
seltnumberin non-circular ducts, leadingto thedevelop-
ment of a model for the Graetz problem in non-circular
ducts. First,itwasobserved[13,16]thattheNusseltnum-
bers for polygonalducts collapsed to asingle point when
L = p
A, see Table 1, for both fully developed and slug
ows. Next it was observedthat when the Nusselt num-
bers for the rectangular duct and elliptic duct, were nor-
malizedwithrespecttothelimitingvaluesforthecircular
and square ducts, these curves were approximately equal
to the normalized friction factor Reynolds numbercurve,
Eq. (30). Finally, amodel wasbeen developed which ac-
curatelypredictsthedataforboththermalboundarycon-
ditions [16], i.e. (T) and (H). The resultingmodel which
requiresEq. (30)is
Nu p
A
=C
1
fR e p
A
8 p
(31)
whereC
1
isequalto 3:01forthe(UWT)boundarycondi-
tion and 3:66 for the (UWF) boundary condition. These
resultsare the average valuefor fully developedow in a
polygonal tube when thecharacteristiclengthscale is the
square root of cross-sectional area[16]. The parameter
is chosen based upon the geometry. Values for which
dene the upper and lower bounds in Figs. 1 and 2 are
xedat =1=10and= 3=10,respectively. Almost all
Eq.(31),withfewexceptions.
Figures 1-3 compare the data for many duct shapes
obtained from Shah and London [1] for both singly and
doubly connectedducts. Whentheresultsarebasedupon
L= p
A, andanappropriateaspectratiodened,twodis-
tinct bounds are formed in Figs. 1 and 2for the Nusselt
number. Thelowerboundconsistsofallductshapeswhich
have re-entrant corners, i.e. angles less than 90 degrees,
while the upper bound consists of all ducts with rounded
cornersand/orrightangledcorners.
Figure 3showstheNusselt numberfor the(H)condi-
tion for polygonalannular ducts and thecircular annular
duct. The predictedvaluesare determined from Eq. (31)
usingtheequivalentdenitionofaspectratiogiveninPart
Iforthecircularannulus. Forthepolygonalannularducts,
the equivalent r
= p
A
i
=A
o
, where A
i and A
o
are the
cross-sectional areas of the inner and outer ducts respec-
tively.
Table 1
NusseltNumbersforSlug and Fully
Developed Flow for Regular Polygons
Isoux(H) Isothermal (T)
Geometry FDF Slug FDF Slug
Triangle 3.11 - 2.47 -
Square 3.61 7.08 2.98 4.93
Nu
D
h
Hexagon 4.00 7.53 3.35 5.38
Octagon 4.21 7.69 3.47 5.53
Circular 4.36 7.96 3.66 5.77
Triangle 3.51 - 2.79 -
Square 3.61 7.08 2.98 4.93
Nu p
A
Hexagon 3.74 7.01 3.12 5.01
Octagon 3.83 7.00 3.16 5.03
Circular 3.86 7.06 3.24 5.11
Short DuctAsymptote,L<<L
t
;L>>L
h
If the velocity distribution is fully developed and the
temperaturedistributionis allowedto develop, theclassic
Graetzproblemresults. Inthethermalentranceregion,the
resultsareweakfunctionsoftheshapeandgeometryofthe
duct. This behaviouris characterizedbythefollowingap-
proximateanalyticalexpressionattributedto Leveque,see
[20]:
Nu/
C
L
1=3
(32)
whereC
isthedimensionlessmeanvelocitygradientatthe
ductwallandL
isthedimensionlessaxiallocation. Thus,
ifC
isaweakfunctionofshape,thenNuwillbeaweaker
function ofshapeduetotheonethirdpower.
In the thermal entrance region of non-circular ducts
the thermal boundary layer is thin and is assumed to be
developinginaregionwherethevelocitygradientislinear.
Forverysmall distances from the duct inlet, the eect of
curvatureontheboundarylayerdevelopmentisnegligible.
Thus, the analysis considers the duct boundary as a at
plate. Thegoverningequationforthissituationisgivenby
Fig. 1 - Fully Developed Flow Nu
T
, data from
Ref.[1].
Fig. 2 - Fully Developed Flow Nu
H
, data from
Ref.[1].
Fig. 3 - Fully Developed Flow Nu
H
in Annular
Ducts,data from Ref. [1].
Cy
@T
@z
=
@ 2
T
@y 2
(33)
where the constant C represents themean velocity gradi-
isdened as:
C=
@u
@n
w
= 1
P I
@u
@n
w
ds (34)
For hydrodynamically fully developed ow, the con-
stant C is related to the friction factor-Reynolds number
product
fR e
L
2
=
@u
@n
w L
U
=C
(35)
where Lisanarbitrarylengthscale.
Deningthefollowingparameters:
Y = y
L Z=
z=L
R e
L Pr
=T T
o R e
L
= UL
leadstothegoverningequation
C
Y
@
@Z
=
@ 2
@Y 2
(36)
Thegoverningequationcannowbetransformedintoan
ordinarydierentialequationforeachwallconditionusing
asimilarityvariable[20]
= Y
(9Z =C
) 1=3
(37)
Both theUWT and UWFconditionsare examined. Solu-
tions to theLevequeproblem are discussed in Bird et al.
[20]. ThesolutionforlocalNusseltnumberwiththeUWT
conditionyields:
Nu
L
= 3
p
3 (2=3)
2
fR e
L
18Z
1=3
=0:4273
fR e
L
Z
1=3
(38)
while the solution for the local Nusselt number for the
(UWF)conditionyields:
Nu
L
= (2=3)
fR e
L
18Z
1=3
=0:5167
fR e
L
Z
1=3
(39)
TheaverageNusseltnumberforbothcasescanbeob-
tainedbyintegratingEqs. (38,39):
Nu
L
= 3
2 Nu
L
(40)
Thesolutionforeachwallconditioncanbecompactly
written withZ=L
as:
Nu
L
=C
2 C
3
fR e
L
L
1=3
(41)
where thevalueofC
2
is1forlocalconditionsand3/2for
averageconditions,andC
3
takesavalueof0.427forUWT
and0.517 forUWF.
The Leveque approximation is valid where the ther-
mal boundary layer is thin, in the region near the wall
where the velocity prole is linear. This corresponds to
owoflargePrandtlnumberuids,whichgiverisetovery
ically fully developed ow. The weak eect of duct ge-
ometry in the entrance region is due to the presence of
the friction factor-Reynoldsnumber product, fR e, in the
aboveexpression,whichisrepresentativeoftheaverageve-
locitygradientat theductwall. ThetypicalrangeoffR e
is 12 < fR e
D
h
< 24, Shah and London [1]. This results
in 2:29 < (fR e
D
h )
1=3
< 2:88, which illustrates the weak
dependency of the thermal entrance region on shape and
aspect ratio. Further reductions are achieved for similar
shapedductsbyusingthelengthscaleL= p
A.
Short Duct Asymptote,L<<L
t
;L<<L
h
Finally, if both hydrodynamic and thermal boundary
layersdevelopsimultaneously, theresultsare strong func-
tionsoftheuidPrandtlnumber.Inthecombinedentrance
regionthebehaviourforverysmallvaluesofL
maybead-
equatelymodelledbytreatingtheductwallasaatplate.
Thecharacteristicsofthisregionare:
Nu
z
p
R e
z
= 8
<
:
0:564Pr 1=2
Pr!0
0:339Pr 1=3
Pr!1
(42)
fortheUWT condition[4],and
Nu
z
p
R e
z
= 8
<
:
0:886Pr 1=2
Pr!0
0:464Pr 1=3
Pr!1
(43)
fortheUWF condition[3].
Composite models for each wall conditionwere devel-
opedbyChurchillandOzoe[3,4]usingtheasymptoticcor-
relation method of Churchilland Usagi [17]. The results
canbedevelopedintermsofthePr!0behaviourorthe
Pr!1behaviour. Forinternalowproblems,theappro-
priateformischosentobein termsofthePr!0charac-
teristicwhichintroducesthePecletnumberPe=R ePr:
Nu
z
(R e
z Pr)
1=2
=
C
o
"
1+
C
o Pr
1=6
C
1
n
#
1=n
=f(Pr) (44)
where C
o and C
1
represent the coeÆcients of the right
handsideofEqs. (42,43). Thecorrelationparameternis
foundby solvingEq. (44)at an intermediate valueof Pr
wheretheexactsolutionisknown,i.e. Pr=1. Thisleads
ton=4:537fortheUWT conditionandn=4:598forthe
UWFcondition. Forsimplicity,n=9=2ischosenforboth
cases.
TheaverageNusselt numberforbothcasesisnowob-
tainedbyintegratingEq. (44):
Nu
L
=2Nu
L
(45)
AfterintroducingL
,thesolutionforeachwallcondi-
tioncanbecompactlywrittenas:
Nu
L
=C
4 f(Pr)
p
L
(46)
4 4
foraverageconditions,and f(Pr)isdenedas:
f(Pr)=
0:564
h
1+ 1:664Pr 1=6
9=2 i
2=9
(47)
fortheUWTcondition,and
f(Pr)=
0:886
h
1+ 1:909Pr 1=6
9=2 i
2=9
(48)
for the UWF condition. The preceding results are valid
onlyforsmallvaluesofL
.
MODELDEVELOPMENTANDCOMPARISONS
A modelwhichisvalidovertheentire rangeofdimen-
sionlessductlengthsforPr!1,wasdevelopedbyMuzy-
chkaandYovanovich[13]by combiningEq. (31)with Eq.
(41) using the Churchill and Usagi [17] asymptotic corre-
lation method. The form of the proposed model for an
arbitrarycharacteristiclengthscaleis:
Nu(z
)= (
C
2 C
3
fR e
z
1
3 )
n
+( Nu
fd )
n
!
1=n
(49)
Now using the result for the fully developed friction
factor,Eq. (32),andtheresultforthefullydevelopedow
Nusselt number, Eq. (33), with n 5 a new model [13]
wasproposedhavingtheform:
Nu p
A (L
)=
2
4 (
C
2 C
3
fR e p
A
L
1
3 )
5
+
C
1
fR e p
A
8 p
5 3
5 1
5
(50)
where the constants C
1
; C
2 , C
3
and are given in Table
2. Theseconstantsdene thevariouscasesforlocal orav-
erage Nusselt number andisothermal orisoux boundary
conditions for the Graetz problem. The constant C
2 was
modiedfromthatfoundbytheLevequeapproximationto
providebetteragreementwiththedata.
A model for thecombinedentranceregion is now de-
velopedbycombiningthesolutionforaatplatewiththe
modelfor theGraetzowproblemdevelopedearlier. The
proposedmodeltakestheform:
Nu p
A (L
;Pr)= 2
4 0
@ (
C
2 C
3
fR e p
A
L
1
3 )
5
+
C
1
fR e p
A
8 p
5
!
m=5
+
C
4 f(Pr)
p
L
m 3
5 1=m
(51)
which is similar to that proposed by Churchill and Ozoe
[3,4]forthecircular duct. Thismodelisacompositesolu-
tionofthethreeasymptoticsolutionsjust presented.
Fig. 4 - Thermally Developing Flow Nu
T;m , data
from Ref.[1].
Fig. 5 - Thermally Developing Flow Nu
H;z , data
from Ref.[1].
Fig. 6 - Thermally Developing Flow in Parallel
Plate Channel, datafrom Ref.[1].
The parameter m was determined to lie in the range
2< m <7, for all data examined. Values for the blend-
ingparameterwerefoundtobeweakfunctionsoftheduct
aspect ratio and whether alocal oraverage Nusselt num-
foundtobemostdependentupontheuidPrandtlnumber.
A simplelinearapproximationwasdeterminedtopro-
vide better accuracy than choosing a single value for all
duct shapes. Due tothevariationin geometries anddata,
higher order approximations oered no additional advan-
tage. Therefore, the linear approximationwhich predicts
theblendingparameterwithin30percentwasfoundto be
satisfactory. Variations in the blending parameter of this
order will lead to small errors in the model predictions,
whereasvariationsontheorderof100percentormore,i.e.
choosing axed value, produce signicantly largererrors.
Theresultingtforthecorrelationparametermis:
m=2:27+1:65Pr 1=3
(52)
The abovemodel is valid for0:1 < Pr <1 which is
typicalformostlowReynoldsnumberowheatexchanger
applications.
Comparisons withtheavailable datafromRef. [1]are
providedinTabularformin[13,14,16]andFigs. 4-9. Good
agreementis obtained with thedata for the circular duct
and parallel plate channel. Note that comparison of the
modelfortheparallelplatechannelwasobtainedbyconsid-
eringarectangularducthavinganaspectratioof=0:01.
Thisrepresentsareasonableapproximationforthissystem.
The data are also compared withthe models of Churchill
and Ozoe[3,4]andStephan[5,6]forthecircular ductand
parallel platechannelin Figs. 7and8.
The numerical data for the UWT circular duct fall
short of the model predictions at low Pr numbers. How-
ever,allof themodelsare inexcellentagreementwiththe
integralformulationofKreith[22]. Goodagreementisalso
obtainedforthecaseofthesquareductforallPrandtlnum-
bers. Comparisonsofthemodelwithdatafortherectangu-
larduct atvarious aspect ratiosandtheequilateraltrian-
gularductshowthatlargerdiscrepanciesarise. Thesedata
neglect theeects oftransverse velocities in boththemo-
mentum and energyequations. Additional graphical com-
parisoncanbefoundin[14].
Theaccuracyforeachcasemaybeimprovedconsider-
ablybyusingtheoptimalvalueoftheparameterm. How-
ever,thisintroducesanadditionalparameterintothemodel
whichisdeemedunnecessaryforpurposesofheatexchanger
design. Theproposedmodelpredictsmostoftheavailable
dataforthecombinedentryproblemtowithin15percent
andmostofthedataforthedataforGraetzowwithin
12percent[13,14,16]. Thesemodelscanalsobeusedtopre-
dicttheheattransfercharacteristicsforothernon-circular
ductsforwhichtherearepresentlynodata.
Thepresentmodelsalsoagreewellwiththepublished
models of Churchill and Ozoe [3,4] and the models of
Stephan[5,6]. ItisevidentfromFigs. 4-9thatthepresent
modelsprovideacceptableaccuracyfordesigncalculations.
Thermal Entrance Lengths
Anequationforpredictingthethermalentrancelength
isnowobtainedfromEq. (29)withB
1
correspondingtoEq.
2 2 3
L
t
=19:80
C
2 C
3
C
1
3
1+3
(1+) 2
1 192
5
tanh
2
2
(53)
Using Eq. (53) with the appropriate values of
C
1
;C
2
;C
3
; yields, L
t
= 0:0358 and L
t
= 0:0366 for
the circular duct for (T) and (H) respectively. When
rescaledto bebased uponthe hydraulicdiameter, we ob-
tain L
t
= 0:0281 and L
t
= 0:0287 for the circular duct
for (T) and (H), which compare with the valuesfrom [1],
L
t
=0:0335andL
t
=0:0430for(T)and(H),respectively.
SUMMARYAND CONCLUSIONS
A general model for predicting the heat transfer co-
eÆcientinthecombinedentryregionofnon-circularducts
was developed. This model is valid for 0:1 < Pr < 1,
0<L
<1,bothuniform walltemperature (UWT)and
uniformwallux(UWF)conditions,andforlocalandmean
Nusseltnumbers. Modelpredictions agreewith numerical
datatowithin15percentorbetterformostnon-circular
ductsandchannels. Themodelwasdevelopedbycombin-
ing theasymptotic resultsof laminarboundarylayerow
andGraetz owforthe thermalentrance region. Inaddi-
tion,bymeansof anovelcharacteristiclength,thesquare
root of cross-sectional area, results for many non-circular
ductsofsimilaraspectratiocollapseontoasinglecurve.
The present study took advantage of scale analysis,
asymptoticanalysis,andtheselectionofamoreappropri-
ate characteristic length scale to developa simple model.
Thismodelonlyrequires twoparameters,theaspect ratio
oftheductandthedimensionlessductlength. Whereasthe
modelsofYilmazandCihan[11,12]forGraetzowconsist
of several equations. The presentmodel predicts mostof
thedevelopingowdata within12percentorbetter.
Finallythesemodelsmayalsobeusedtopredictresults
forductsforwhichnosolutionsortabulateddataexist.
ACKNOWLEDGMENTS
Theauthors acknowledgethe nancialsupport of the
Natural Sciences and Engineering Research Council of
Canada(NSERC).
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Table2
CoeÆcientsfor GeneralModel
Boundary Condition
UWT(T) C1=3:01; C3=0:409 f(Pr)=
0:564
h
1+(1:664Pr 1=6
) 9=2
i
2=9
UWF(H) C1=3:66; C3=0:501 f(Pr)=
0:886
h
1+(1:909Pr 1=6
) 9=2
i
2=9
NusseltNumberType
Local C
2
=1 C
4
=1
Average C2=3=2 C4=2
ShapeParameter
UpperBound =1=10
LowerBound = 3=10
