Laminar Flow Friction and Heat Transfer in Non- Circular Ducts and Channels Part I-

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Laminar Flow Friction and Heat Transfer in Non- Circular Ducts and Channels Part I-

Hydrodynamic 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 123-130

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IN NON-CIRCULAR DUCTS AND CHANNELS

PART I - HYDRODYNAMIC 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

Adetailedreviewandanalysisofthehydrodynamiccharacteristicsoflaminardevelopingandfullydevelopedowinnon-

circularductsispresented. NewmodelsareproposedwhichsimplifythepredictionofthefrictionfactorReynoldsproduct

fR efordevelopingandfullydevelopedowinmostnon-circularductgeometriesfoundinheatexchangerapplications. By

meansofscalinganalysisitisshownthatcompleteproblemmaybeeasilyanalyzedbycombiningtheasymptoticresultsfor

theshortandlongduct. Throughtheintroductionofanewcharacteristic lengthscale,thesquarerootofcross-sectional

area, the eectofduct shapehas beenminimized. The newmodelhasanaccuracyof 10percent or betterfor most

commonductshapes. Bothsinglyanddoublyconnectedductsareconsidered.

NOMENCLATURE

A = owarea,m

2

a;b = majorandminor axesof ellipseorrectangle,m

C

1

;C

2

= constants

c = lineardimension,m

D = diameterofcircularduct,m

D

h

= hydraulicdiameter,4A=P

E() = completeellipticintegralofthesecondkind

e = eccentricity,m

e

= dimensionlesseccentricity,e=(r

o r

i )

f = frictionfactor=(

1

2 U

2

)

K = incremental pressuredropfactor

L = ductlength,m

L +

= dimensionlessductlength,L=LR e

L

N = numberofsidesofapolygon

n = correlationparameter

P = perimeter, m

p = pressure,N=m 2

r = radius,m

r

= dimensionlessradiusratio,r

i

=r

o

R e

L

= Reynoldsnumber,UL=

~

V = velocityvector,m=s

u;v;w = velocitycomponents,m=s

U = averagevelocity,m=s

x;y;z = cartesiancoordinates,m

Greek Symbols

Æ = boundarylayerthickness,m

= aspectratio,b=a

= halfangle, rad

= dynamicviscosity,Ns=m 2

= uiddensity,kg=m 3

= wallshearstress, N=m 2

Subscripts

p

A = baseduponthesquarerootofowarea

c = core

D

h

= baseduponthehydraulicdiameter

h = hydrodynamic

i = inner

L = baseduponthearbitrarylengthL

o = outer

1 = fullydevelopedlimit

Superscripts

C = circle

E = ellipse

P = polygon

R = rectangle

INTRODUCTION

Laminar ow uid friction and heat transfer in non-

circularductsoccursquitefrequentlyinlowReynoldsnum-

berowheatexchangerssuchascompactheatexchangers.

Itisnowalsooccurringmorefrequentlyinelectroniccooling

applicationsasaresultoftheminiaturizationofpackaging

technologies. Whiletraditionalapproachesrelyheavilyon

the use of tabulated and/or graphicaldata, theability to

designthermalsystemsusing robustmodelsismuch more

desirable. Most modern uid dynamics and heat transfer

texts rarely present correlations ormodels for more com-

plexgeometrieswhichappearinmanyengineeringsystems.

Rather,asubsetofdataformiscellaneousgeometriesisusu-

allypresentedafterdetaileddiscussionandanalysisofsim-

ple geometriessuch asthe circularductand parallel plate

channel.

Intherstpartofthispaper,thehydrodynamicprob-

lem is considered in detail, andanew model is developed

forpredictingthefrictionfactorReynoldsnumberproduct

for developing laminar ow in non-circular ducts. In the

second partofthis paper,theassociated thermalproblem

is considered,and new models are developed forthe both

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binedentranceproblem.

Laminarfullydevelopeduidowinnon-circularducts

ofconstantcross-sectionalarearesultswhentheductlength

issuÆciently greater thanthe entrancelength, L >>L

h ,

or when the characteristic transversal scale is suÆciently

small to ensure a small Reynolds number. Under these

conditionstheowthroughmostof theductmaybecon-

sideredfullydeveloped. Inmanyengineeringsystemssuch

ascompactheatexchangersandmicro-coolersusedinelec-

tronicspackaging, thecharacteristicdimensionoftheow

channelissmallenoughtogiverisetofullydevelopedow

conditions. However, in many modern systems, the ow

lengthisgenerallynotverylarge,LL

h

orL<<L

h ,and

developing ow prevails over most of the duct length. In

these situations, amodel capable ofpredicting thehydro-

dynamiccharacteristic,usuallydenotedasfR e,thefriction

factorReynoldsnumberproduct,isrequired.

A review of the literature reveals that only two sig-

nicantattempts atdevelopingageneralmodelhavebeen

undertaken. ThesearetheworkofShah[1]andYilmaz[2].

ThesemodelsarebasedupontheearlyworkofBender[3].

Bender[3]combinedtheasymptoticresultofShapiroetal.

[4],withtheresultforthe\long"duct,to provideamodel

whichisvalidovertheentirelengthofacircularduct. Shah

[1]laterextendedthemodelofBender[3]topredictresults

forthe equilateraltriangle, the circularannulus, therect-

angularduct, and parallelplate channel geometries. Shah

[1]achieved this bygeneralizingthe form of the model of

Bender [3], and tabulating coeÆcientsfor each particular

case. Recently, Yilmaz[2]proposedamoregeneralmodel

ofShah[1]. RatherthantabulatingcoeÆcients,Yilmaz[2]

developed a complex correlation scheme for the fully de-

veloped frictionfactorfR e,theincremental pressuredrop

K

1

, andattingcoeÆcientC whichappearsintheShah

[1] model. This model is moregeneral than that of Shah

[1]but isalsoquitecomplex.

Despite itscomplexity, the model of Yilmaz [2]is ac-

curate over theentirerange of the entranceand fully de-

velopedregionsformanyductcross-sections. Theprimary

drawbackofthe simplemodelproposedbyShah [1]is the

requirement of tabulated coeÆcients and parameters for

each geometry, i.e. fR e, K

1

, and C, thus limiting in-

terpolationforgeometriessuchastherectangularductand

circular annulus, whose solutionvaries with aspect ratio.

Inthecaseof themodeldevelopedbyYilmaz [2],interpo-

lation isno longeraproblem, however,this isachievedat

thecostofsimplicity.

Thetwomodelsdiscussedaboverepresentthecurrent

stateoftheartforinternalowproblems. Bothmodelsare

baseduponthecombinationofthe\short"ductand\long"

ductsolutionsusing thecorrelatingmethod of Bender[3].

InthisapproachtheincrementalpressuredropfactorK

1

isrequired in the\long duct"solution. As aresultof the

complexcorrelatingequationsforK

1

developedbyYilmaz

[2],thesimplephysicalbehaviourofthehydrodynamicen-

trance problem is lost. It is apparent from the available

data, that smooth transition occurs from theentrancere-

obtainedby Shapiroet al. [4]accountsforthe increasein

momentumoftheacceleratingcore,useofthetermK

1 in

ahydrodynamic entrancemodelsuchasthat proposed by

Bender[3]isredundant. Themodelpresentedinthispaper

doesnotrequirethisparameterandissignicantlysimpler

in structure.

GOVERNING EQUATIONS

Thegoverningequationsforsteadyincompressibleow

inthehydrodynamicentranceregioninanon-circularduct

orchannelare:

r

~

V =0 (1)

~

V r

~

V = rp+r 2

~

V (2)

Simultaneous solution to the contiuity, Eq. (1), and

momentum,Eq. (2),equationssubjecttothe noslipcon-

dictionattheductwall,

~

V =0,theboundednesscondition

along the duct duct axis,

~

V 6= 1, and a constant initial

velocity,

~

V =U

~

k, arerequiredtocharacterizetheow. In

thenextsection,scalinganalysisisusedtoshowtheappro-

priate form of the solutionfor both short and longducts.

Later,asymptoticanalysisisusedtodevelopanewmodel.

SCALE ANALYSIS

Wenowexaminethemomentumequationandconsider

the various force balances implied under particular ow

conditions. Themomentum equationrepresentsabalance

ofthree forces: inertia,pressure,andfriction,i.e.

~

V r

~

V

| {z }

Inertia

= rp

|{z}

Pressure

+ r

2

~

V

| {z }

Friction

(3)

We nowconsider three separate force balances. Each

isexaminedbelowusingthemethodofscaleanalysisadvo-

catedbyBejan[5].

Long Duct Asymptote,L>>L

h

Fullydevelopedlaminarowinaductofarbitrary,but

constantcross-section,isgovernedbythePoissonequation:

rp

|{z}

Pressure

= r

2

~

V

| {z }

Friction

(4)

which represents abalance betweenthe viscous and pres-

sureforces. Thuswemaywritethefollowingapproximate

relationusingthecharacteristicsoftheowandthegeom-

etry:

p

L

U

L 2

(5)

whereLrepresentsacharacteristictransversallengthscale

of theduct cross-section. Thevelocity scalesaccordingto

theareameanvalue,U,andtheaxiallengthscalesaccord-

ingto L. Rearrangingtheaboveexpressiongives:

p

UL

L 2

(6)

Next, we examine the shear stress at the wall. The

shear stressmaybeapproximatedby

(4)

~ =r

~

V

U

L

(7)

Next, we may introduce the denition of the friction

factordenedasthedimensionlesswallshear:

f

U 2

U=L

U 2

1

R e

L

(8)

Theaboveexpressionmaybewrittensuchthatthefol-

lowingrelationshipexistsforallcross-sectionalgeometries:

fR e

L

U

O(1) (9)

or

fR e

L

=C

1

(10)

TheconstantC

1

hasbeenfound to vary formost geome-

triesin therange 12<C

1

<24, whenL=D

h

. Finally, a

controlvolumeforcebalancegives

= p

L A

P

(11)

yields the following relationship when combined with the

twoscalinglaws,Eqs. (6,7):

L A

P

D

h

4

(12)

Weshall see later that this length scalewhich results

fromtheforcebalance,althoughconvenient,isnotthemost

appropriatechoice.

Short Duct Asymptote,L<<L

h

Intheentranceregion nearthe ductinlet tworegions

mustbeconsidered. Oneistheinviscidcoreandtheother

theviscousboundarylayer. Theowwithintheboundary

layer is very similar to laminar boundary layer develop-

mentoveraatplate,exceptthatthevelocityattheedge

oftheboundarylayerisnotconstant. Thetworegionsare

nowexaminedbeginningwiththeowwithintheboundary

layer.

The force balance within the boundary layer is gov-

ernedstrictly byinertia andfrictionforces:

~

V r

~

V

| {z }

Inertia

= r

2

~

V

| {z }

Friction

(13)

Thisbalanceyields:

U

2

L

U

Æ 2

(14)

whereÆistheboundarylayerthickness. Thus,

Æ

L

1

p

R e

L

(15)

Next,considering therelationshipforthewallshear:

~ =r

~

V

U

Æ

(16)

andthefrictionfactor,weobtain:

f

U 2

U=Æ

U 2

R e

L

UL

1

p

R e

L

(17)

Re-writingtheaboveexpressionintermsofthelength

scale L, and dening the product of friction factor and

Reynolds number yields the following expression for the

entranceregion:

fR e

L

O(1)

p

L +

(18)

or

fR e

L

= C

2

p

L +

(19)

where

L +

= L

LR e

L

(20)

isthedimensionlessductlength.

Finally, in the inviscid core the momentum equation

representsabalanceofinertia andpressureforces:

~

V r

~

V

| {z }

Inertia

= rp

|{z}

Pressure

(21)

whichscalesaccordingto

U 2

p (22)

Since the pressureat any point alongthe duct in the

developingregionmustbeconstantin thecoreandin the

boundary layer,and using the force balance given by Eq.

(11),itis notdiÆculttosee thattheconstantinEq. (19)

will bemuch largerthanthe valueobtainedfor boundary

layerowoveraatplate. TheconstantC

2

hasbeenfound

theoretically [4] for the circular duct to beC

2

=3:44 for

a mean friction factor and C

2

= 1:72 for a local friction

factor.

In summary, we havefound from scaling analysis the

followingrelationshipsforthefrictionfactorReynoldsnum-

berproduct:

fR e

L

= 8

>

<

>

: C

1

L>>L

h

C

2

p

L +

L<<L

h

(23)

This asymptotic behaviour will be examined further

and will form the basis for the new model developed for

the hydrodynamicentranceproblem. Finally, anequation

relating the approximate magnitude of the hydrodynamic

entrancelengthmaybeobtainedbyconsideringanequality

between thetwoasymptoticlimits givenin Eq. (23)with

L +

=L +

h :

C

1

= C

2

q

L +

h

(24)

or

L +

h

=

C

2

C

1

2

(25)

(5)

pareswellwith moreexactsolutions. Thissolutionrepre-

sentstheintersection oftheasymptoticlimitsonaplotof

thecompletebehaviouroffR eversusL +

.

ASYMPTOTIC ANALYSIS

Inthissection,asymptoticanalysis[6]isusedtoestab-

lish expressions for the characteristic longduct and short

ductbehaviourestablishedthroughscalinganalysis. First,

the longduct limit is considered and asimple expression

developedforpredictingtheconstantfR e=C

1

. Addition-

ally, theissueofanappropriatecharacteristiclengthscale

isaddressed. Finally, theshortductlimitisconsidered by

re-examiningtheapproximate solutionobtainedby Siegel

[7].

Long DuctAsymptote,L>>L

h

In order to establish the longduct limit, solutionsto

Eq. (4)formanyductsareexamined. Thesesolutionshave

beencataloguedin[8,9]. Thesimplest ductshapesarethe

circularductand theparallel plate channel. These impor-

tantshapesalsoappearaslimitsin theellipticalduct, the

rectangularduct,andthecircularannulus.

Oneissue which hasnot been addressedin thelitera-

tureistheselectionofanappropriatecharacteristiclength

scale,i.e. L. Traditionally,thehydraulicdiameterhasbeen

chosen,L=4A=P. However,inmanytextsitsuseinlami-

narowhasbeenquestioned[10-12]. Inanearlierwork[13],

the authors addressedthis issue using dimensional analy-

sis. Itwasdeterminedthatthewidelyused conceptofthe

hydraulicdiameter wasinappropriateforlaminarowand

the authors proposed using L = p

A as a characteristic

lengthscale,byconsidering otherproblemsin mathemati-

calphysicsforwhichthePoissonequationapplies. Amore

detaileddiscussionandanalysisontheuseofL= p

Amay

befoundin Muzychka[14].

We will now examine a number of important results

employingbothL=4A=P andL= p

A. Startingwiththe

elliptical duct, the dimensionless average wall shear, Eq.

(9),isfoundtobe[8]:

fR e

D

h

=

2(1+ 2

)

E(

0

)

(26)

where = b=a, the ratio of minor and major axes, and

0

= p

1

2

. Equation(26)hasthefollowinglimits:

fR e

Dh

= 8

<

:

16 !1

19:76 !0

(27)

IfthesolutionisrecastusingL= p

A, asacharacter-

isticlengthscale,thefollowingrelationshipisobtained:

fR e p

A

= 2

3=2

(1+ 2

)

p

E(

0

)

(28)

Equation(28)hasthefollowinglimits:

fR e p

A

= 8

>

<

>

: 8

p

=14:18 !1

2 3=2

p

= 11:14

p

!0

(29)

for the dimensionless average wall shear [8], considering

onlythersttermoftheseriesgives:

fR e

D

h

=

24

(1+ 2

)

1 192

5

tanh

2

(30)

fR e

Dh

= 8

<

:

14:13 !1

24 !0

(31)

Examinationofthesingletermsolutionrevealsthatthe

greatesterroroccurs when=1,whichgivesafR evalue

0.7 percentbelowtheexactvalueoffR e=14:23. Results

for awiderangeofaspectratiosaretabulatedin [8]using

athirtytermserieswhichprovidedsevendigitprecision.

IfthesolutionisrecastusingL= p

A, asacharacter-

isticlengthscale,thefollowingexpressionis obtained:

fR e p

A

=

12

p

(1+)

1 192

5

tanh

2

(32)

fR e p

A

= 8

>

<

>

:

14:13 !1

12

p

!0

(33)

Table 1presentsa comparison of the exact values[8]

withthesingletermapproximation,Eq. (30)andEq. (32)

for both characteristic length scales. Also presented are

valueswhichresultfrom usingtheasymptoticsolutionfor

theparallel platechannel. Clearly,thisasymyptoticresult

doesanadequatejobofpredictingthevaluesoffR e p

A up

to =0:7. Beyond this aspect ratio, verylittle change is

observedinthefR evalues.

Table 1

Comparisonof Single Term

Approximation forfR e

fR eD

h

fR e p

A

=b=a Exact Eq.(30) Exact Eq. (32) 12 = p

0.001 23.97 23.97 379.33 379.33 379.47

0.01 23.68 23.68 119.56 119.56 120.00

0.05 22.48 22.47 52.77 52.77 53.66

0.1 21.17 21.16 36.82 36.81 37.95

0.2 19.07 19.06 25.59 25.57 26.83

0.3 17.51 17.49 20.78 20.76 21.91

0.4 16.37 16.34 18.12 18.09 18.97

0.5 15.55 15.51 16.49 16.46 16.97

0.6 14.98 14.94 15.47 15.43 15.49

0.7 14.61 14.55 14.84 14.79 14.34

0.8 14.38 14.31 14.47 14.40 13.42

0.9 14.26 14.18 14.28 14.20 12.65

1 14.23 14.13 14.23 14.13 12.00

(6)

rectangularductsolutions. Itisnowapparentthattheso-

lutionforthecircularductandsquareducthaveessentially

collapsed toa singlevalue, Eqs. (29,33). Further,in the

limitofsmall aspect ratio,theresultsfor theellipticduct

and the rectangular duct have also come closer together,

Eqs. (29, 33). It is also clearfrom this analysis,that the

squarerootofcross-sectionalareaismoreappropriatethan

thehydraulic diameter for non-dimensionalizing thelami-

narowdata. AsseeninTable2,themaximumdierence

between the values for fR e occur in the limit of ! 0.

Comparison of Eq. (29) and Eq. (33) shows this dier-

ence is only 7.7 percent. Thus, we may use the simpler

expression, Eq. (32), to compute valuesfor the elliptical

duct. This way, the elliptic integralin Eq. (28)need not

beevaluated.

Table2

fR eResultsfor Ellipticaland Rectangular

Geometries [8]

fR eD

h

fR e p

A

=b=a Rect. Ellip.

fR e R

fR e E

Rect. Ellip.

fR e R

fR e E

0.01 23.67 19.73 1.200 119.56 111.35 1.074

0.05 22.48 19.60 1.147 52.77 49.69 1.062

0.10 21.17 19.31 1.096 36.82 35.01 1.052

0.20 19.07 18.60 1.025 25.59 24.65 1.038

0.30 17.51 17.90 0.978 20.78 20.21 1.028

0.40 16.37 17.29 0.947 18.12 17.75 1.021

0.50 15.55 16.82 0.924 16.49 16.26 1.014

0.60 14.98 16.48 0.909 15.47 15.32 1.010

0.70 14.61 16.24 0.900 14.84 14.74 1.007

0.80 14.38 16.10 0.893 14.47 14.40 1.005

0.90 14.26 16.02 0.890 14.28 14.23 1.004

1.00 14.23 16.00 0.889 14.23 14.18 1.004

Table3

fR eResultsfor Polygonal Geometries[8]

N fR eD

h fR e

P

fR e C

fR e p

A fR e

P

fR e C

3 13.33 0.833 15.19 1.071

4 14.23 0.889 14.23 1.004

5 14.73 0.921 14.04 0.990

6 15.05 0.941 14.01 0.988

7 15.31 0.957 14.05 0.991

8 15.41 0.963 14.03 0.989

9 15.52 0.970 14.04 0.990

10 15.60 0.975 14.06 0.992

20 15.88 0.993 14.13 0.996

1 16 1.000 14.18 1.000

Next, we consider the regular polygons. Values for

fR e

Dh

fallin therange13:33fR e

Dh

16for3N

1. Therelativedierencebetweenthetriangularand the

circular ducts is approximately 16.7 percent. When the

characteristiclengthscaleischangedtoL= p

A , therela-

tivedierenceis reducedto7.1 percent fortheequilateral

gons. TheresultsaresummarizedinTable3. Solutionsfor

anumberofothercommongeometriesareshowninFigs.1

and2asafunctionofaspectratio. Thedenitionofaspect

ratioissummarizedinTable4foranumberofgeometries.

Withtheexceptionofthetrapezoidandtheannularsector,

the aspect ratio for all singlyconnected ducts is taken as

the ratioofthemaximum widthto maximumlengthsuch

that 0 < < 1. For the trapezoid, annular sector, and

the doubly connected duct, simple expressions have been

devised to relatethe characteristicdimensionsof theduct

to awidthto lengthratio.

Fig. 1- fR e p

A

for Regular FlatDucts,

Data from Ref. [8]

Fig. 2- fR e p

A

for Other Non-Circular Ducts,

Data from Ref. [8]

Thenal resultsto beconsidered arethose ofthecir-

cularannulusand otherannularductswhich arebounded

externally by apolygonorinternally by apolygon [8]. It

is clear from Fig. 3 that excellent agreement is obtained

when theresults arerescaledaccordingto L = p

A anda

newaspectratiodened asr

= p

A

i

=A

o

. This denition

waschosensinceitreturnsthesamer

ratioforthecircular

annularduct. ValuesforfR eforthecircularannulusand

other shapeshavebeenexaminedbyMuzychka[14].

(7)

Fig. 3 -fR e p

A

forDoubly ConnectedDucts,

Data from Ref. [8]

Table4

Denitionsof Aspect Ratio

Geometry AspectRatio

RegularPolygons =1

Singly-Connected = b

a

Trapezoid =

2b

a+c

AnnularSector = 1 r

(1+r

)

CircularAnnulus = (1 r

)

(1+r

)

EccentricAnnulus = (1+e

)(1 r

)

(1+r

)

It should be noted that as the inner boundary ap-

proachestheouterboundary,thereissomedeparturefrom

thecircular annulus resultdue to theoweld becoming

multiplyconnected. Thesepointshavenotbeenshownon

theplot. Limiting valuesof r

areprovided in [13,14]. It

hasbeenfound that Eq. (32)maybeused to predict the

resultsofthecircularannulusprovidedthefollowingequiv-

alentsinglyconnectedaspectratioisdened:

= (1 r

)

(1+r

)

(34)

Thisresultmaybeobtainedfrom twodierentphysi-

calarguments. Therst istheratiooftheofgap,r

o r

i ,

tothemeanperimeter,whilethesecondisobtainedasthe

ratio ofthe gap to the equivalent lengthif the duct area,

(r 2

o r

2

i

),isconvertedtoarectangle. Bothpointsofview

yieldthesamedenition.

It is now clear that Eq. (32) fully characterizes the

owinthelongductlimit. Themaximumdeviationofex-

act values is of the order 7-10 percent. It has now been

shownthatthedimensionlessaveragewallshear,fR e,may

tionoftheaspectratioischosen.

Short DuctAsymptote,L<<L

h

An analytical result for the friction factor in the en-

trance region of the circular duct was obtainedby Siegel

[7] using several methods. The solution begins with the

denition of the short duct friction factor which may be

obtainedbywritingBernoulli'sequationintheentrancere-

gion. Sincethepressuregradientisonlyafunctionofaxial

position, the following relationship may be written which

relatesthefrictionfactortothevelocityintheinviscidcore

p

i p

L

1

2 U

2

=

u

c

U

2

1=4f

L

D

(35)

Inordertodeterminethefrictionfactor,arelationship

forthedimensionlesscorevelocityneedstobefound. Siegel

[7] applied several approximate analyticalmethods to ob-

tain a solution for the velocity in the inviscid core. The

most accurate method wasthe application of theMethod

of Thwaites, (see Goldstein [15]). The Siegel [7] analysis

beginswiththeintegratedformofthecontinuityequation

in the entance region where the boundary layer is small

relativetotheductdiameter:

4 D

2

U =

4

(D 2Æ) 2

u

c +

Z

Æ

0

(D 2y)udy (36)

where uisthevelocitydistributionin theboundarylayer,

u

c

isthe velocity in thecore, and U is themean velocity.

Rearrangingthisexpression leadsto

U

u

c

=1 4

D Z

Æ

0

1 u

u

c

dy+

8

D 2

Z

Æ

0

1 u

u

c

ydy (37)

Next, using Pohlhausen's approximate velocity dis-

tribution, Siegel [7] developed an expression relating the

boundary layer displacement thickness to the velocity in

the core. Siegel [7] then obtainedthe followingfour term

approximationforthevelocityinthecoreneartheentrance

ofacircularduct

u

c

U

=1+6:88(L +

) 1=2

43:5(L +

)+1060(L +

) 3=2

(38)

Substitution oftheaboveresultintotheexpressionfor

thefrictionfactoryields:

fR e

D

= 3:44

(L +

) 1=2

h

1 2:88(L +

) 1=2

+111(L +

) i

(39)

AtkinsonandGoldstein[15]obtainedasolutionforthe

corevelocityusingamethodproposedbyShiller[16]which

solvesforthevelocityin thecoreusing aseriesexpansion.

TheresultsofAtkinsonandGoldstein[15]yieldsimilarre-

sults, withtheleadingtermintheseriesbeingexactlythe

same. Asimilaranalysisfortheparallelchannel[16]yields

thesameleadingtermasthatforthecircularduct. Analy-

sisoftheexpressionsdevelopedbySiegel[7],andAtkinson

(8)

non-dimensionalized using any characteristic length scale

withoutintroductionofscalingterms:

fR e

L

= 3:44

(L +

) 1=2

"

1 2:88(L +

) 1=2

L

D

+111(L +

)

L

D

2

#

(40)

whereL +

isdened byEq. (20).

However,rescalingtheadditionaltermsin theexpres-

sionresultsinscaling parameterswhichare nowfunctions

oftheductgeometry. Thusveryneartheinletofanynon-

circularduct, theleadingtermofthesolutionisvalid. As

theboundarylayerbeginstogrowfurtherdownstream,the

eectsofgeometrybecomemorepronouncedandthesolu-

tion for the circular duct is no longer valid. The leading

terminthesolutionforanycharacteristiclengthLis

fR e

L

= 3:44

p

L +

(41)

whichisvalidforL +

=L=LR e

L

0:001.Ifalocalfriction

factorisdesired,theconstant3.44isreplacedby1.72.

Equation (41) is independent of the duct shape and

may be used to compute the friction factor for the short

ductasymptoteofmostnon-circularducts.

MODELDEVELOPMENTANDCOMPARISONS

A general model is now proposed using the Churchill

andUsagi [17]asymptoticcorrelationmethod. Themodel

takestheform:

fR e p

A

=

( C

1 )

n

+

C

2

p

L +

n

1=n

(42)

where nisasuperposition parameterdetermined by com-

parisonwithnumericaldataoverthefullrangeofL +

. Us-

ing the results provided by Eq. (32) and Eq. (41), and

the general expression, Eq. (42), the following model is

proposed:

fR e p

A

=

2

6

4 0

B

@

12

p

(1+)

1 192

5

tanh

2

1

C

A 2

+

3:44

p

L +

2 3

7

5 1=2

(43)

Usingtheavailabledata[8,9],itisfoundthatthevalue

of nwhich minimizesthe root meansquare dierence lies

in the range 1:5 < n < 3:6 with a mean value n 2,

[14,18]. Twenty-six data sets were examined from Refs.

[8,9] and are summarized in [14,18]. Comparisons of the

modelarepresentedin Figs4-6forthemostcommonduct

shapes. With the exceptionof the eccentricannularduct

atlargevaluesofr

ande

,i.e. acrescentshape,thepro-

posed model predictsallof thedevelopingowdata avail-

ablein theliteraturetowithin 10percentor betterwith

fewexceptions. Theproposedmodelprovidesequalorbet-

teraccuracythanthemodelofYilmaz[2]andisalsomuch

simpler. A comparisonof themodel withthedatafor the

try p

A !1, however, this geometry may be accurately

modeledasarectangular duct with =0:01or acircular

annular duct with r

>0:5. Good agreementis obtained

with the currentmodel whenthe parallel plate channel is

modeledasaniteareaductwithsmallaspect ratio.

One notable feature of the new model is that it does

notcontaintheincrementalpressuredroptermK

1 which

appearsin themodelsof Bender[3], Shah[1], andYilmaz

[3]. Since thesolutionof Siegel [7]fortheentranceregion

accounts for both the wall shear and the increase in mo-

mentum due to the acceleratingcore, there is no need to

introducethe termK

1

. Thus theproposedmodelis now

only a function of the dimensionless duct length L +

and

aspect ratio,whereasthemodelsofShah [1]andYilmaz

[2]arefunctionsof manymoreparameters.

Finally,itisdesirabletodevelopanexpressionforthe

hydrodynamic entrance length. Traditionally, the length

of hydrodynamic boundary layer development in straight

ducts ofconstantcross-sectionalareaisusually dened as

the point wherethe centerlinevelocityis 0:99U

max , Shah

andLondon[8]. Anewdenitionforthehydrodynamicen-

trancelengthmaybeobtainedfromEq. (25). Substituting

Eq. (32)forC

1 andC

2

=3:44,anapproximateexpression

fortheentrancelengthisobtained:

L +

h

=0:0822(1+) 2

1 192

5

tanh

2

(44)

The expression above reduces to L +

h

= 0:059 when

= 1 and L +

h

= 0:00083 when = 0:01. These lengths

when rescaledto bebasedonthe hydraulic diameter take

the valuesL +

h

=0:047 when =1and L +

h

=0:021 when

= 0:01. They compare with the results from Shah and

London [8]for thecircular duct, L +

h

=0:056,and parallel

plate channel,L +

h

=0:011.

SUMMARY ANDCONCLUSIONS

A simplemodelwasdeveloped forpredictingthe fric-

tionfactorReynoldsnumberproductinnon-circularducts

fordevelopinglaminarow. Thepresentstudytookadvan-

tageofscaleanalysis,asymptoticanalysis,andtheselection

ofamoreappropriatecharacteristiclengthscaletodevelop

asimplemodel. Thismodelonlyrequirestwoparameters,

the aspect ratio of the duct and the dimensionless duct

length. Whereas themodel of Shah [1]requires tabulated

valuesofthreeparameters,andthemodelofYilmaz[2]con-

sistsofseveralequations. Thepresentmodelpredictsmost

of the developingowdata within 10 percent orbetter

for8singlyconnectedductsand2doublyconnectedducts.

Finally this model may also be used topredict resultsfor

ductsforwhichnosolutionsortabulateddataexist. Itwas

also shownthat thesquareroot ofthecross-sectionalow

area wasa more eective characteristic length scale than

thehydraulicdiameterforcollapsingthenumericalresults

ofgeometrieshavingsimilarshapeandaspectratio.

ACKNOWLEDGMENTS

The authors acknowledge thenancial support of the

Natural Sciences and Engineering Research Council of

Canada(NSERC).

(9)

Fig. 4 - fR e p

A

for Developing Laminar Flow in

Polygonal Ducts,Data from Ref. [8]

Fig. 5 - fR e p

A

for Developing Laminar Flow in

Regular FlatDucts,Data from Ref. [8]

Fig. 6-fR e p

A

forDevelopingLaminarFlow inthe

CircularAnnularDuct, Data from Ref. [8]

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dynamic Entry Length Solutions for Circular and Non-

Circular Ducts", Journalof Fluids Engineering, Vol. 100,

1978,pp. 177-179.

[2] Yilmaz, T., \GeneralEquationsfor PressureDrop for

LaminarFlowinDuctsofArbitraryCrossSections",Jour-

nal of Energy Resources Technology, Vol. 112, 1990, pp.

220-223.

[3] Bender,E.,\DruckverlustbeiLaminarerStromungim

Rohreinlauf", Chemie-Ingenieur Technik, Vol. 41, 1969,

pp. 682-686.

[4] Shapiro, A.H., Siegel, R., and Kline, S.J., \ Friction

Factor in the Laminar Entry Region of a Smooth Tube",

Proceedings of the 2nd U.S. National Congress of Applied

Mechanics,1954,pp. 733-741.

[5] Bejan, A., Convection Heat Transfer, 2 nd

Ed., Wiley,

1995,NewYork,NY.

[6] Leal, L.G., Laminar Flow and Convective Transport,

Butterworth-Heinemann,1992,Boston,MA.

[7] Siegel, R., The Eect of Heating on Boundary Layer

Transition for LiquidFLow ina Tube,Sc.D.Thesis,1953,

MassachussetsInstituteof Technology,Boston,MA.

[8] Shah, R.K. and London, A.L., Laminar Flow Forced

ConvectioninDucts,AcademicPress,1978,NewYork,NY.

[9]Shah,R.K.andBhatti,M.S.,Chapter 3: LaminarCon-

vectiveHeatTransferinDucts,inHandbookofSinglePhase

Convective Heat Transfer, eds. S. Kakac, R.K. Shah and

W. Aung,Wiley,1987,NewYork,NY.

[10] White, F.M.,ViscousFluid Flow,McGraw-Hill,1974,

NewYork,NY.

[11] Denn, M., Process Fluid Mechanics, Prentice-Hall,

1980,UpperSaddleRiver,NJ.

[12]Mironer,A.FluidMechanics,McGraw-Hill,1979,New

York,NY.

[13] Yovanovich, M.M.and Muzychka, Y.S., \Solutions of

PoissonEquationwithinSinglyandDoublyConnectedDo-

mains",AIAA97-3880,1997 NationalHeatTransferCon-

ference,August10-12,1997,BaltimoreMD.

[14]Muzychka,Y.S.,Analytical andExperimentalStudy of

FluidFrictionandHeatTransferinLowReynoldsNumber

Flow HeatExchangers,Ph.D.Thesis,UniversityofWater-

loo,1999,Waterloo,ON.

[15]Goldstein,R.(ed.),ModernDevelopmentsinFluidDy-

namics,Oxford,1938,Oxford,UK.

[16]Schlichting,H.,BoundaryLayerTheory,McGraw-Hill,

1979,NewYork,NY.

[17]Churchill,S.W.andUsagi,R.,\AGeneralExpression

fortheCorrelationofRatesofTransferandOtherPhenom-

ena", American Instituteof Chemical Engineers, Vol. 18,

1972,pp. 1121-1128.

[18] Muzychka, Y.S. and Yovanovich, M.M., \Modelling

FrictionFactorsinNon-CircularDuctsforDevelopingLam-

inar Flow", AIAA Paper 98-2492, 2 nd

AIAA Theoertical

Fluid Mechanics Meeting, June15-18, 1998,Albuquerque,

NM.

Laminar Flow Friction and Heat Transfer in Non- Circular Ducts and Channels Part I-