Analysis and optimization of the thermal performance of ...dli9/Publications/IJNMHF...

Cooling Technologies Research Center

Research Publications

Purdue Libraries Year

Analysis and optimization of the thermal

performance of microchannel heat sinks

Dong Liulowast S V Garimella dagger

lowast

daggerPurdue Univ sureshgpurdueedu

This paper is posted at Purdue e-Pubs

httpdocslibpurdueeducoolingpubs9

Analysis and optimizationof the thermal performance of

microchannel heat sinksDong Liu and Suresh V Garimella

Cooling Technologies Research Center School of Mechanical EngineeringPurdue University West Lafayette Indiana USA

Abstract

Purpose ndash To provide modeling approaches of increasing levels of complexity for the analysis ofconvective heat transfer in microchannels which offer adequate descriptions of the thermalperformance while allowing easier manipulation of microchannel geometries for the purpose of designoptimization of microchannel heat sinks

Designmethodologyapproach ndash A detailed computational fluid dynamics model is first used toobtain baseline results against which five approximate analytical approaches are compared Theseapproaches include a 1D resistance model a fin approach two fin-liquid coupled models and a porousmedium approach A modified thermal boundary condition is proposed to correctly characterize theheat flux distribution

Findings ndash The results obtained demonstrate that the models developed offer sufficiently accuratepredictions for practical designs while at the same time being quite straightforward to use

Research limitationsimplications ndash The analysis is based on a single microchannel while in apractical microchannel heat sink multiple channels are employed in parallel Therefore theoptimization should take into account the impact of inletoutlet headers Also a prescribed pumpingpower may be used as the design constraint instead of pressure head

Practical implications ndash Very useful design methodologies for practical design of microchannelheat sinks

Originalityvalue ndash Closed-form solutions from five analytical models are derived in a format thatcan be easily implemented in optimization procedures for minimizing the thermal resistance ofmicrochannel heat sinks

Keywords Optimization techniques Heat transfer Convection

Paper type Technical paper

The Emerald Research Register for this journal is available at The current issue and full text archive of this journal is available at

wwwemeraldinsightcomresearchregister wwwemeraldinsightcom0961-5539htm

NomenclatureAc frac14 microchannel cross-sectional areaAf frac14 fin cross-sectional areaAs frac14 area of heat sinkCp frac14 specific heatDh frac14 hydraulic diameterf Re frac14 friction constant

h frac14 heat transfer coefficientH frac14 height of heat sinkHc frac14 microchannel depthk frac14 thermal conductivityL frac14 length of heat sink_m frac14 mass flow rate

The authors acknowledge the financial support from members of the Cooling TechnologiesResearch Center (available at httpwidgetecnpurdueedu CTRC) a National ScienceFoundation IndustryUniversity Cooperative Research Center at Purdue University

Analysis andoptimization

7

Received February 2003Revised November 2003

Accepted May 2004

International Journal for NumericalMethods in Heat amp Fluid Flow

Vol 15 No 1 2005pp 7-26

q Emerald Group Publishing Limited0961-5539

DOI 10110809615530510571921

IntroductionThe potential for handling ultra-high heat fluxes has spurred intensive research intomicrochannel heat sinks (Tuckerman and Pease 1981 Weisberg and Bau 1992Sobhan and Garimella 2001) For implementation in practical designs the convectiveheat transfer in microchannels must be analyzed in conjunction with the choice andoptimization of the heat sink dimensions to ensure the required thermal performanceDesign procedures are also needed to minimize the overall thermal resistance

The focus of this paper is to present a comprehensive discussion and comparison offive different (approximate) analytical models of increasing sophistication which offerclosed-form solutions for single phase convective heat transfer in microchannelsA general computational fluid dynamics (CFD) model is first set-up to obtain anldquoexactrdquo solution Details of the approximate models and the assumptions involved arethen presented along with a comparison of the thermal resistance predictions fromthese models Optimization of the thermal performance of microchannel heat sinks isthen discussed

Description of the problemThe microchannel heat sink under consideration is shown in Figure 1 Materials forfabrication may include conductive materials such as copper and aluminumfor modular heat sinks or silicon if the microchannels are to be integrated into thechip For conservative estimates of thermal performance the lid (top plate) may be

n frac14 number of microchannelsNu frac14 Nusselt numberP frac14 pressureq00 frac14 heat fluxq frac14 heat removal rateQ frac14 volume flow rateR frac14 thermal resistanceRe frac14 Reynolds numbert frac14 substrate thicknessT frac14 fin temperatureTb frac14 temperature at the base of the finTf frac14 fluid temperatureum frac14 mean flow velocityW frac14 width of heat sinkwc frac14 microchannel widthww frac14 fin thickness

Greek symbolsa frac14 aspect ratio of microchannelshf frac14 fin efficiencym frac14 dynamic viscosityu frac14 thermal resistancer frac14 density of fluidDP frac14 pressure dropDT frac14 temperature difference

Subscripts and superscriptsc frac14 channel fluidf frac14 fluidi frac14 inlets frac14 solid finw frac14 wall

Figure 1Schematic of amicrochannel heat sink

HFF151

8

to be insulated The width of individual microchannels and intervening fins (wcthornww)is typically small compared to the overall heat sink dimension W and numerouschannels are accommodated in parallel flow paths

Continuum equations for conservation of mass momentum and energyrespectively for the convective heat transfer in microchannel heat sinks can bewritten as (Fedorov and Viskanta 2000 Toh et al 2002)

7 middot ethr kVTHORN frac14 0 eth1THORN

kV middot7ethr kVTHORN frac14 27P thorn 7 middot ethm7 kVTHORN eth2THORN

kV middot7ethrCpTTHORN frac14 7 middot ethkf7TTHORN for the fluid eth3THORN

7 middot ethks7TsTHORN frac14 0 for the fin eth4THORN

This set of equations assumes steady-state conditions for incompressible laminarflow with radiation heat transfer neglected With an appropriate set of boundaryconditions these equations provide a complete description of the conjugate heattransfer problem in microchannels

CFD modelA numerical model was formulated to solve the three-dimensional heat transfer inmicrochannels using the commercial CFD software package FLUENT (Fluent Inc1998) The simulation was performed for three different sets of dimensions as listed inTable I These three cases are chosen to simulate experiments in the literature(Tuckerman and Pease 1981) that have often been used for validating numericalstudies (Weisberg and Bau 1992 Toh et al 2002 Ryu et al 2002)

The computational domain chosen from symmetry considerations is shown inFigure 2 The top surface is adiabatic and the left and right sides are designatedsymmetric boundary conditions A uniform heat flux is applied at the bottomsurface In the present work water is used as the working fluid (r frac14 997 kg=m3Cp frac14 4 179 J=kg K m frac14 0000855 kg=ms and kf frac14 0613 W=mK evaluated at 278C)and silicon is used as the heat sink substrate material with ks frac14 148 W=mK

In the numerical solution the convective terms were discretized using afirst-order upwind scheme for all equations The entire computational domain wasdiscretized using a 500 pound 60 pound 14 (x-y-z) grid To verify the grid independence of

Case1 2 3

wc (mm) 56 55 50ww (mm) 44 45 50Hc (mm) 320 287 302H (mm) 533 430 458DP (kPa) 10342 11721 21373q00 (Wcm2) 181 277 790Rexp (8CW) (Tuckerman and Pease 1981) 0110 0113 0090Rnum (8CW) 0115 0114 0093

Note L frac14 W frac14 1 cm

Table IComparison of thermal

resistances


9

the convective heat transfer results three different meshes were used in the fluidpart of the domain 20 pound 5 30 pound 7 and 50 pound 15 The thermal resistance changed by34 percent from the first to the second mesh and only by 03 percent upon furtherrefinement to the third grid Hence 30 pound 7 grids were used in the fluid domain forthe results in this work

The agreement between the experimental and predicted values of thermalresistance in Table I validates the use of the numerical predictions as abaseline against which to compare the approximate approaches considered in thiswork

The numerical results may also be used to shed light on the appropriate boundaryconditions for the problem under consideration For instance it is often assumed inmicrochannel heat sink analyses that the axial conduction in both the solid fin andfluid may be neglected Using the numerical results for case 1 as an example the axialconduction through the fin and fluid were found to account for 03 and 02 percent ofthe total heat input at the base of the heat sink respectively Thus the assumption ofnegligible axial conduction appears valid for heat transfer in the silicon microchannelsconsidered

Two alternative boundary conditions have been commonly used at the base of thefin in microchannel analyses (Zhao and Lu 2002 Samalam 1989 Sabry 2001)

2ksrsaquoT

rsaquoy

yfrac140

frac14 q00 eth5THORN

or

2ksdT

dy

yfrac140

frac14wc thorn ww

wwq00 eth6THORN

in which equation (5) implies that the imposed heat flows evenly into the fluid via thebottom of the microchannel and into the fin via the base of the fin while equation (6)implies that all the heat from the base travels up the base of the fin Clearly neither of

Figure 2Computational domain

HFF151

10

these two extreme cases represent the actual situation correctly The computed heatflux in the substrate in the immediate vicinity of the fin base is shown in Figure 3 forcase 1 The heat fluxes into the fluid and the fin are 555 and 333 Wcm2 respectivelyHence the error associated with employing equations (5) and (6) as the boundarycondition at the base of the fin would be 50 and 24 percent respectively A reasonablyaccurate alternative for the boundary condition could be developed as follows

q frac14 hwc

2L

ethTb 2 T fTHORN thorn hethH cLTHORNhfethTb 2 T fTHORN eth7THORN

Hence the ratio of the heat dissipated through the vertical sides of the fin to thatflowing through the bottom surface of the microchannel into the fluid is 2hfHcwc or

2hfa This leads to a more reasonable boundary condition at the base of the fin

2ksdT

dy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww

wwq00 eth8THORN

This condition results in a heat flux of 366 Wcm2 through the base of the fin which iswithin 10 percent of the computed exact value of 333 Wcm2

In light of this discussion equation (8) is imposed as the thermal boundarycondition at the base of the fin for all the five approximate models developed in thiswork

Approximate analytical modelsIn view of the complexity and computational expense of a full CFD approach forpredicting convective heat transfer in microchannel heat sinks especially in searchingfor optimal configurations under practical design constraints simplified modelingapproaches are sought The goal is to account for the important physics even if some

Figure 3Heat flux distribution at

the base of the fin


11

of the details may need to be sacrificed Five approximate analytical models (Zhao andLu 2002 Samalam 1989 Sabry 2001 Kim and Kim 1999) are discussed along withthe associated optimization procedures needed to minimize the thermal resistanceThe focus in this discussion is on the development of a set of thermal resistanceformulae that can be used for comparison between models as well as for optimizationof microchannel heat sinks

As shown in Figure 1 for the problem under consideration the fluid flows parallelto the x-axis The bottom surface of the heat sink is exposed to a constant heat fluxThe top surface remains adiabatic

The overall thermal resistance is defined as

Ro frac14DTmax

q00Aseth9THORN

where DTmax frac14 ethTwo 2 T fiTHORN is the maximum temperature rise in the heat sink ie thetemperature difference between the peak temperature in the heat sink at the outlet(Two) and the fluid inlet temperature (Tfi) Since the thermal resistance due tosubstrate conduction is simply

Rcond frac14t

ksethLW THORNeth10THORN

the thermal resistance R calculated in following models will not include this term

R frac14 Ro 2 Rcond eth11THORN

The following assumptions are made for the most simplified analysis

(1) steady-state flow and heat transfer

(2) incompressible laminar flow

(3) negligible radiation heat transfer

(4) constant fluid properties

(5) fully developed conditions (hydrodynamic and thermal)

(6) negligible axial heat conduction in the substrate and the fluid and

(7) averaged convective heat transfer coefficient h for the cross section

In the approximate analyses considered this set of assumptions is progressivelyrelaxed

Model 1 ndash 1D resistance analysisIn addition to making assumptions 1 ndash 7 above the temperature is assumed uniformover any cross section in the simplest of the models

For fully developed flow under a constant heat flux the temperature profile withinthe microchannel in the axial direction is shown in Figure 4 The three components ofthe heat transfer process are

qcond frac14 ksAsTwo 2 Tbo

teth12THORN

HFF151

12

qconv frac14 hAfethTb 2 T fTHORN eth13THORN

qcal frac14 rQCpethT fo 2 T fiTHORN eth14THORN

The overall thermal resistance can thus be divided into three components

Ro frac14DTmax

q00ethLW THORNfrac14

1

q00ethLW THORNfrac12ethTwo 2 T fiTHORN frac14 Rcond thorn Rconv thorn Rcal eth15THORN

in which the three resistances may be determined as follows

(1) Conductive thermal resistance

Rcond frac14t


(2) Convective thermal resistance

Rconv frac141

nhLeth2hfH c thorn wcTHORNeth17THORN

with fin efficiency hf frac14 tanhethmH cTHORN=mH c

(3) Caloric thermal resistance

Rcal frac141

rfOCpeth18THORN

Model 2 ndash fin analysisIn this model assumptions 1-7 mentioned above are adopted and the fluid temperatureprofile is considered one-dimensional (averaged over y-z cross section) T f frac14 T fethxTHORNThe temperature distribution in the solid fin is then

Figure 4Temperature profile in a

microchannel heat sink


13

d2T

dy 2frac14

2h

kswwethT 2 T fethxTHORNTHORN eth19THORN

with boundary conditions

2ksdT

dy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww

wwq00 eth20THORN

dT

dy

yfrac14H c

frac14 0 eth21THORN

It follows that

Tethx yTHORN frac14 T fethxTHORN thorn1

m

q00

ks

2hfa

2hfathorn 1

wc thorn ww

ww

coshmethH c 2 yTHORN

sinhethmH cTHORNeth22THORN

where m frac14 eth2h=kswwTHORN1=2

The fluid temperature Tf(x) can be obtained from an energy balance

_mCpdT fethxTHORN

dxfrac14 q00ethwc thorn wwTHORN eth23THORN

with T fethx frac14 0THORN frac14 T0 The bulk fluid temperature is then

T fethxTHORN frac14 T0 thornq00ethwc thorn wwTHORN

rfCpumH cwcx eth24THORN

and equation (22) can be rewritten as

Tethx yTHORN frac14 T0 thorn1

m

q00

ks

2hfa

2hfathorn 1

wc thorn ww

ww


sinhethmH cTHORNthorn

q00ethwc thorn wwTHORN


The thermal resistance is thus

R frac14DT


TethL 0THORN2 T0

q00ethLW THORN

frac141

m

1

ks

2hfa

2hfathorn 1

wc thorn ww

ww

coshethmH cTHORN

sinhethmH cTHORN

1

ethLW THORNthorn

ethwc thorn wwTHORN

rfCpumH cwc

1

Weth26THORN

Model 3 ndash fin-fluid coupled approach IFollowing the same line of reasoning as in the fin analysis (model 2) and adoptingassumptions 1-7 mentioned above but averaging the fluid temperature only in the zdirection (Samalam 1989) the energy equation in the fin can be written as

rsaquo2T

rsaquoy 2frac14

2h

kswwethT 2 T fethx yTHORNTHORN eth27THORN

with

HFF151

14

2ksrsaquoT

rsaquoy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww

wwq00 frac14 J eth28THORN

rsaquoT

rsaquoy

yfrac14H c

frac14 0 eth29THORN

The energy balance in the fluid is represented by

rfCpumwcrsaquoT f

rsaquoxfrac14 2hethT 2 T fethx yTHORNTHORN eth30THORN

and it is assumed that T fethx frac14 0 yTHORN frac14 0 Substituting h frac14 Nukf=Dh into equation (30)yields

1

2rfCpumwcDh

rsaquoT f

rsaquoxthorn kfNuT f frac14 kfNuT eth31THORN

Defining X frac14 x=a and Y frac14 y=a where a frac14 rfCpumwcDh=2kfNu the solution toequation (31) can be written as

T fethX Y THORN frac14

Z X

0

TethX 0Y THORNe2ethX2X 0THORN dX 0 eth32THORN

Hence equation (27) can then be transformed to

rsaquo2TethX Y THORN

rsaquoY 2frac14 b TethX Y THORN2

Z X

0

TethX 0Y THORNe2ethX2X 0THORN dX 0

eth33THORN

in which

b frac14a2

l 2eth34THORN

l 2 frac14kswwDh

2kfNueth35THORN

Solving equation (33) by Laplace transforms

rsaquo2f ethY sTHORN

rsaquoY 2frac14 gf g frac14

bs

s thorn 1

eth36THORN

The boundary conditions in equations (28) and (29) become

2ksrsaquof

rsaquoY

Yfrac140

frac14Ja

seth37THORN

2ksrsaquof

rsaquoY

Yfrac14 ~Hc

frac14 0 eth38THORN

where ~Hc frac14 H c=a and J is defined in equation (28) The solution to this system ofequations is


15

f ethsTHORN frac14Ja

kssffiffiffig

pcosh

ffiffiffig

pethY 2 ~HcTHORN

sinh

ffiffiffig

p ~Hc

eth39THORN

The inverse Laplace transform yields the temperature

TethX Y THORN frac14 L21frac12f ethsY THORN

frac14Ja

ks~Hcb

eth1 thorn XTHORN thornb

2ethY 2 ~HcTHORN

2 2b ~H

2

c

6

(

thorn 2X1nfrac141

eth21THORNnethsn thorn 1THORN2

sncos

npethY 2 ~HcTHORN

~Hc

esnX

)eth40THORN

in which

sn frac142n 2p 2= ~H

2

c

bthorn n 2p 2= ~H2

c

This is a rapidly converging infinite series for which the first three terms adequatelyrepresent the thermal resistance

R frac14DT


TethL 0THORN2 T0


Ja

ks~Hcb

eth1 thorn L=aTHORN thorn1

3b ~H

2

c

1

ethLW THORNeth41THORN

Model 4 ndash fin-fluid coupled approach IIIn this model assumptions 1-7 mentioned above are again adopted except that axialconduction in the fin is not neglected (Sabry 2001) The governing equations in thesolid fin and liquid respectively are therefore

72Tethx y zTHORN frac14 0 eth42THORN

7 middot ethrfCp kVT fethx y zTHORNTHORN frac14 kf72T fethx y zTHORN eth43THORN

At the fin-fluid interface the condition is

2ksrsaquoTs

rsaquozfrac14 2kf

rsaquoT f

rsaquozfrac14 hethT i 2 T fTHORN eth44THORN

in which the averaged local fluid temperature is

T fethxTHORN frac14

Z wc=2

0

vT f dz=ethumwc=2THORN eth45THORN

with

HFF151

16

um frac14

Z wc=2

0

v dz=wc=2

Along with equation (28) the following boundary conditions apply

rsaquoT

rsaquoy

yfrac14H c

frac14rsaquoT

rsaquox

xfrac140

frac14rsaquoT

rsaquox

xfrac14L

frac14rsaquoT

rsaquoz

zfrac142ww

2

frac14rsaquoT f

rsaquoz

zfrac14wc

2

frac14 0 eth46THORN

Integrating equation (42) over z from 2ww2 to 0 the fin temperature varies as

ww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2

T thorn

rsaquoT

rsaquoz

0

2ww2

frac14 0 eth47THORN

in which

T frac14

Z 0

2ww2

T dz=ethww=2THORN

Combining equations (44) and (46)

ksww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2

T 2 hethTi 2 T fTHORN frac14 0 eth48THORN

Assuming Ti frac14 T since Bi frac14 hethww=2THORN=ks 1 equation (48) becomes

ksww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2

T 2 hethT 2 T fTHORN frac14 0 eth49THORN

If the axial conduction term rsaquo2T=rsaquox 2 is neglected equation (49) would reduce toequation (27)

Since fully developed conditions are assumed and axial conduction in the fluid isneglected equation (43) may be integrated over z from 0 to wc2 to yield

rsaquo

rsaquox

Z wc=2

0

uT f dz frac14kf

rfCp

rsaquoT f

rsaquoz

wc2

0

eth50THORN

Using the boundary condition in equation (44) this reduces to

umwc

2

rsaquo

rsaquoxT f thorn

1

rfCphethT f 2 TTHORN frac14 0 eth51THORN

The following dimensionless variables are introduced

X frac14 x=L Y frac14 y=H c and T frac14T 2 T0

DTceth52THORN

in which


17

DTc frac142hfa

2hfathorn 1

ethwc thorn wwTHORN=2

hH cq00 eth53THORN

The system of equations above can be cast in dimensionless terms

A 2 rsaquo2

rsaquoX 2thorn

rsaquo2

rsaquoY 2

T 2 ethmH cTHORN

2ethT 2 T fTHORN frac14 0 eth54THORN

rsaquoT f

rsaquoXthorn SethT f 2 TTHORN frac14 0 eth55THORN

rsaquoT

rsaquoY

Yfrac140

frac14 2ethmH cTHORN2 eth56THORN

rsaquoTs

rsaquoX

Xfrac141

frac14rsaquoTs

rsaquoY

Yfrac140

frac14rsaquoTs

rsaquoY

Yfrac141

frac14 0 eth57THORN

T fjXfrac140 frac14 0 eth58THORN

where A frac14 H c=L and the modified Stanton number (Sabry 2001) is given by

S frac14 hL

rfCpumwc

2

Employing similar techniques as adopted for model 3 the fin temperature isobtained as

TethX Y THORN frac14 mH ccoshethmH ceth1 2 Y THORNTHORN

sinhethmH cTHORNthornX1nfrac140

cosethnpY THORNf nethXTHORN eth59THORN

in which the first term in the infinite series provides results of acceptable accuracy(5 percent deviation from the complete series)

f 0ethXTHORN frac14 SX thornX2

ifrac141

C0i

w0i

ew0iX thorn C03 eth60THORN

with

w01 frac14 2S

2thorn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS

2

2

thornmH c

A

2s

w02 frac14 2S

22


2

2

thornmH c

A

2s

C01 frac14 2Sethew02 2 1THORN

ethew02 2 ew01 THORN

HFF151

18



C03 frac14SA

mH c

2

The thermal resistance is thus obtained as

R frac14DT


Teth1 0THORN2 T0

q00ethLW THORNDTc

frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN

sinhethmH cTHORNthorn S thorn

X2

ifrac141

C0i

w0i

ew0i thorn C03

1

ethLW THORN

eth61THORN

In most practical cases ethmH c=ATHORN S=2 and equation (61) reduces to

R frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


SA

mH c

2

1


Model 5 ndash porous medium approachThe convective heat transfer process in microchannels can also be treated as beingsimilar to that in a fluid-saturated porous medium with the extended Darcy equationused for fluid flow and a volume-averaged two-equation model used for heat transferas demonstrated in Vafai and Tien (1981)

Following the analysis of Kim and Kim (1999) a two-equation model can beemployed to obtain the volume-averaged properties over a representative elementaryvolume for the solid region and the fluid region separately The momentum equationand boundary conditions are

2d

dxkplf thorn mf

d2

dy 2kulf 2

mf

K1kulf frac14 0 eth63THORN

kulf frac14 0 at y frac14 0H c eth64THORN

where kulf is the volume-averaged velocity 1 frac14 wc=ethwc thorn wwTHORN is the porosity andK frac14 1w2

c=12 is the permeability Equations (63) and (64) may be written as

U frac14 Dad2U

dY 22 P eth65THORN

U frac14 0 at Y frac14 0 1 eth66THORN

using the dimensionless parameters

U frac14kulf

um Da frac14

K

1H 2frac14

1

12a2s

Y frac14y

H P frac14

K

1mfum

dkplf

dx

The solution to the momentum equation is then


19

U frac14 P cosh

ffiffiffiffiffiffiffiffiffiffiffi1

DaY

r thorn

1 2 coshffiffiffiffi1

Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 2 1

8gtltgt

9gt=gt eth67THORN

The volume-averaged energy equations for the fin and fluid respectively are

ksersaquo2kTlrsaquoy 2

frac14 haethkTl2 kTlfTHORN eth68THORN

1rfCpkulf

rsaquokTlf

rsaquoxfrac14 haethkTl2 kTlfTHORN thorn kfe

rsaquo2kTlf

rsaquoy 2eth69THORN


kTl frac14 kTlf frac14 Tw at y frac14 0 eth70THORN

rsaquokTlrsaquoy

frac14rsaquokTlf

rsaquoyfrac14 0 at y frac14 H c eth71THORN

where a is the wetted area per unit volume h the local heat transfer coefficient definedas the ratio of the interfacial heat flux to the solid-fluid temperature difference and kse

and kfe the effective conductivities of the solid and fluid defined as kse frac14eth1 2 1THORNks kfe frac14 1kf

For fully developed flow under constant heat flux it is known that

rsaquokTlf

rsaquoxfrac14

rsaquokTlrsaquox

frac14dTw

dxfrac14 constant eth72THORN

and

q00 frac14 1rfCpumHrsaquokTlf

rsaquoxeth73THORN

The energy equations (68) and (69) and boundary conditions can thus be written indimensionless form as

d2u

dY 2frac14 Dethu2 ufTHORN eth74THORN

U frac14 Dethu2 ufTHORN thorn Cd2uf

dY 2eth75THORN

with

u frac14 uf frac14 0 at Y frac14 0 eth76THORN

du

dYfrac14

duf

dYfrac14 0 at Y frac14 1 eth77THORN

in which

HFF151

20

C frac141kf

eth1 2 1THORNks D frac14

haH 2

eth1 2 1THORNks u frac14

kTl2 Tw

q00Heth121THORNks

uf frac14kTlf 2 Tw

q00Heth121THORNks

Substituting the solution obtained for velocity equations (74) and (75) can be solvedto give

uf frac14P

1 thorn C

242

1

2Y 2 thorn C1Y thorn C2 2 C3 cosh

0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Deth1 thorn CTHORN

CY

r 1A

2 C4 sinh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN

CY

r

thorn C5 cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt375

eth78THORN

u frac14 P

24Da cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt

21

2Y 2 thorn C1Y 2 Da

352 Cuf

eth79THORN

where

N 1 frac14 Deth1 thorn CTHORN

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

Da1 2 cosh

ffiffiffiffiffiffi1

Da

r ( )vuut

N 2 frac14C

Da

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN

C

rsinh

ffiffiffiffiffiffi1

Da

r sinh


C

r

C1 frac14 1 2

ffiffiffiffiffiffiDa

pcosh

ffiffiffiffi1

Da

q 2 1

sinh

ffiffiffiffi1

Da

q

C2 frac14 2Da thorn1

Deth1 thorn CTHORN

C3 frac14C

DaDeth1 thorn CTHORND1


21

C4 frac14N 1 thorn N 2

Deth1 thorn CTHORN

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN

Ccosh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN

CY

q sinh

ffiffiffiffi1

Da

q D1

s

C5 frac14 Da 21

D1

Finally the thermal resistance can be obtained as

R frac1412K

rfCpw3caW1 2um

2ufbH c

eth1 2 1THORNksLWeth80THORN

in which ufb is the bulk mean fluid temperature defined as

uf b frac14

Z 1

0

Uuf dY

13Z 1

0

U dy

Key features of the five approximate models discussed above including theassumptions governing equations and resistance formulae developed are summarizedin Table II

Assessment of the approximate modelsFor the microchannel parameters listed in Table I thermal resistances were computedwith Fluent as well as from the five approximate models The results are shown inTable III It can be seen that all the approximate models would provide acceptablepredictions for the thermal resistance of the microchannel heat sink with the maximumdeviation being 78 percent Models 2-5 are more complex to apply than model 1 andinvolve the solution of the differential governing equations In spite of its simplicity model1 appears to adequately represent the physics of the heat transfer problem and isrecommended for use in the design and optimization of practical microchannel heat sinks

It may be noted that in model 2 the fluid temperature is considered to be only afunction of the x-coordinate and the fin temperature is solved in a truly 1D mannerThe thermal resistance expression from model 2 is therefore identical to that frommodel 1 Also in models 3 and 4 since 2D temperature fields are considered in both thefin and fluid the new terms H c=eth3nkswwLTHORN and eth1=ethrfCpQTHORN2THORNethnksH cww=LTHORN appearin addition to the other terms in the simpler models 1 and 2 The difference betweenmodels 3 and 4 is that the axial conduction term appears explicitly in the fin equationof model 4 while it is neglected in model 3

In the calculations above expressions for Nusselt number Nu and the frictionconstant f Re are needed for computing the convective heat transfer coefficient h andthe average velocity um in the microchannel In all the five approximate modelsdiscussed above the flow is assumed to be thermally and hydrodynamically fullydeveloped Hence the following relations are used in terms of microchannel aspectratios (Incropera and DeWitt 1996 Shah and London 1978)

Nufd frac14 8235eth1 2 1883athorn 3767a 2 2 5814a 3 thorn 5361a 4 2 2a 5THORN eth81THORN

ethf ReTHORNfd frac14 96eth1213553=athorn19467=a 2 217012=a3 thorn09564=a 4 202537=a 5THORN eth82THORN

HFF151

22

Tem

per

atu

reA

xia

lco

nd

uct

ion

Mod

elF

inF

luid

Fin

Flu

idG

over

nin

geq

uat

ion

sT

her

mal

resi

stan

ce(R

)

11D

1Dpound

poundN

otu

sed

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfQ

Cp

(88)

22D

1Dpound

pound

d2T

dy

22

hP

k sA

cethT

2T

fethxTHORNTHORNfrac14

0

_ mC

pdT

fethxTHORN

dx

frac14q

middotethw

cthorn

wwTHORN

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfC

pQ

(89)

32D

2Dpound

poundk srsaquo

2T

rsaquoy

2frac14

2h wwethT

2T

fethx

yTHORNTHORN

rfC

pu

mw

crsaquoT

f

rsaquox

frac142hethT

2T

fethx

yTHORNTHORN

Ja

k s~ H

cb

eth1thorn

L=aTHORNthorn

1 3b~ H

2 c

1

ethLW

THORN(9

0)

42D

2Dp

pound7

2TethxyzTHORNfrac14

0

7middotethk VT

fethx

yzTHORNTHORNfrac14

af7

2T

fethx

yzTHORN

2hfa

2hfa

thorn1

1

2nhH

ch

fLthorn

1

rfC

pQthorn

1

ethrfC

pQTHORN2

nk s

Hcw

w

L

(91)

51D

1Dpound

pound

2d dxkpl fthorn

mf

d2

dy

2kul f2

mf k1kul ffrac14

0

k sersaquo

2kT

lrsaquoy

2frac14

haethk

Tl2

kTl fTHORN

1r

fCpkul frsaquokT

l frsaquox

frac14haethk

Tl2

kTl fTHORNthorn

k fersaquo

2kT

l frsaquoy

2

12K

rfC

pw

3 ca

W1

2u

m2

uf

hH

c

eth12

1THORNk

sL

W(9

2)

Notespound

ndashn

otco

nsi

der

ed

andp

ndashco

nsi

der

ed

Table IISummary of approximate

analytical models


23

However the fully developed assumption is not always valid especially for microchannelswith the larger hydraulic diameters and short lengths With hydrodynamic and thermallengths defined as Lthorn frac14 L=ethDhReTHORN and L frac14 L=ethDhRePrTHORN the following relations(Samalam 1989 Harms et al 1999) could be employed instead of equations (81) and (82)

Nu frac14 335ethL THORN2013a 012Pr20038 0013 L 01 eth83THORN


f appRe frac1432

ethLthornTHORN057

2

thornethf ReTHORN2fd

1=2

Lthorn 005 eth85THORN

In the present calculations L=Dh lt 100 with moderate Reynolds numbers so that thehydrodynamically fully developed condition is satisfied For the working fluid in thisstudy (water Prandtl number 58) Nusselt numbers calculated from equations (81) and(83) are listed in Table IV The deviation between the two sets of results is within 6 percentand therefore the assumption of thermally fully developed conditions is acceptableIn general developing thermal effects should be carefully considered before fullydeveloped conditions are assumed

OptimizationThe optimization of microchannel heat sink design can be motivated using the thermalresistance approach in model 1 As indicated in equation (18) Rcal is inverselyproportional to the mass flow rate When the pressure head along the microchannellength is prescribed as the constraint Rcal will decrease as wc increases when Hc

reaches the maximum allowable value However the convective heat transfercoefficient h will increase when Dh decreases leading to a reduction in Rconv asshown by equation (17) The heat transfer from the substrate through the fins willalso be enhanced if the fin efficiency increases which requires a larger fin

Case

Thermal resistance (8CW) 1 2 3

Ronum 0115 0114 0093Romodel 1 0112 0112 0091Romodel 2 0112 0112 0091Romodel 3 0106 0106 0087Romodel 4 0106 0106 0087Romodel 5 0115 0106 0089

Table IIIOverall thermalresistances

Case1 2 3

Nufd 597 581 606Nu 560 555 585

Table IVNusselt numbers

HFF151

24

thickness ww However the increase in ww will reduce the number of microchannelfinpairs in a heat sink for a prescribed heat sink size Due to these competing factors thereexists an optimal microchannel dimension that minimizes the overall thermalresistance

In order to optimize the thermal performance of a microchannel heat sink thefollowing variables must be specified from implementation constraints

(1) thermal conductivity of the bulk material used to construct the heat sink (ks)

(2) overall dimension of the heat sink (L and W from the size of the chip Hc and tfrom fabrication and structural considerations)

(3) properties of the coolant (rf m kf Cp) and

(4) allowable pressure head (DP)

To illustrate the procedure the example considered uses water as the working fluid tocool a chip with L frac14 W frac14 1 cm and a given pressure head of DP frac14 60 kPa The heatload is 100 Wcm2 The microchannel heat sink is to be made of silicon witht frac14 100mm and H c frac14 400mm The fluid properties are evaluated at 27 8C Theoptimization process involves finding the optimal microchannel geometry (channelwidth wc fin thickness ww and aspect ratio a frac14 H c=wc) that will minimize thermalresistance

Solutions to the following equations would yield the optimum

rsaquoR

rsaquowcfrac14 0 eth86THORN

rsaquoR

rsaquowwfrac14 0 eth87THORN

In this work the optimization computations were performed using the commercialsolver MATLAB (The Math Works Inc 2001) The optimized results derived from thefive approximate models are listed in Table V The optimal thermal resistance valuesreported from the five models agree to the within 10 percent It may also be noted thatthe minimum thermal resistance is always attained at the largest allowable aspectratio In practical designs the aspect ratio would be determined by the limits on themicrochannel depth and the substrate thickness

ConclusionsFive approximate analytical models for predicting the convective heat transfer inmicrochannel heat sinks are presented and compared Closed-form solutions from thesemodels are compared to full CFD simulation and experimental results and the efficacy

Model wc (mm) ww (mm) a Ro (8CW)

1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072

Table VOptimal dimensions


25

of the different models assessed Optimization procedures are discussed for minimizingthe thermal resistance of the heat sinks The results obtained demonstrate that themodels developed offer sufficiently accurate predictions for practical designs while atthe same time being quite straightforward to use

References

Fedorov AG and Viskanta R (2000) ldquoThree-dimensional conjugate heat transfer in themicrochannel heat sink for electronic packagingrdquo International Journal of Heat and MassTransfer Vol 43 pp 399-415

Fluent Inc (1998) Fluent Userrsquos Guide Lebanon New Hampshire

Harms TM Kazmierczak MJ and Gerner FM (1999) ldquoDeveloping convective heat transfer indeep rectangular microchannelsrdquo International Journal of Heat and Fluid Flow Vol 20pp 149-57

Incropera FP and DeWitt DP (1996) Fundamentals of Heat and Mass Transfer WileyNew York NY

Kim SJ and Kim D (1999) ldquoForced convection in microstructure for electronic equipmentcoolingrdquo Journal of Heat Transfer Vol 121 pp 639-45

The MathWorks Inc (2001) Matlab Version 61 Natwick MA

Ryu JH Choi DH and Kim SJ (2002) ldquoNumerical optimization of the thermal performance ofa microchannel heat sinkrdquo International Journal of Heat and Mass Transfer Vol 45pp 2823-7

Sabry M-N (2001) ldquoTransverse temperature gradient effect on fin efficiency for microchanneldesignrdquo Journal of Electronic Packaging Vol 123 pp 344-50

Samalam VK (1989) ldquoConvective heat transfer in microchannelsrdquo Journal of ElectronicMaterials Vol 18 pp 6111-7

Shah RK and London AL (1978) ldquoLaminar flow forced convection in ductsrdquo Advances in HeatTransfer Supplement 1 Academic Press London

Sobhan CB and Garimella SV (2001) ldquoA comparative analysis of studies on heat transfer andfluid flow in microchannelsrdquo Microscale Thermophysical Engineering Vol 5 pp 293-311

Toh KC Chen XY and Chai JC (2002) ldquoNumerical computation of fluid flow and heattransfer in microchannelsrdquo International Journal of Heat and Mass Transfer Vol 45pp 5133-41

Tuckerman DB and Pease RF (1981) ldquoHigh-performance heat sinking for VLSIrdquo IEEEElectronic Devices Letters EDL-2 pp 126-9

Vafai K and Tien CL (1981) ldquoBoundary and inertia effects on flow and heat transfer in porousmediardquo International Journal of Heat and Mass Transfer Vol 24 pp 195-203

Weisberg A and Bau HH (1992) ldquoAnalysis of microchannels for integrated coolingrdquoInternational Journal of Heat and Mass Transfer Vol 35 pp 2465-74

Zhao CY and Lu TJ (2002) ldquoAnalysis of microchannel heat sinks for electronics coolingrdquoInternational Journal of Heat and Mass Transfer Vol 45 pp 4857-69

Further reading

Tien CL and Kuo SM (1987) ldquoAnalysis of forced convection in microstructures for electronicsystem coolingrdquo Proc Int Symp Cooling Tech for Electronic Equipment Honolulu HIpp 217-26

HFF151

26

Analysis and optimizationof the thermal performance of

microchannel heat sinksDong Liu and Suresh V Garimella

Cooling Technologies Research Center School of Mechanical EngineeringPurdue University West Lafayette Indiana USA

Abstract

Purpose ndash To provide modeling approaches of increasing levels of complexity for the analysis ofconvective heat transfer in microchannels which offer adequate descriptions of the thermalperformance while allowing easier manipulation of microchannel geometries for the purpose of designoptimization of microchannel heat sinks

Designmethodologyapproach ndash A detailed computational fluid dynamics model is first used toobtain baseline results against which five approximate analytical approaches are compared Theseapproaches include a 1D resistance model a fin approach two fin-liquid coupled models and a porousmedium approach A modified thermal boundary condition is proposed to correctly characterize theheat flux distribution

Findings ndash The results obtained demonstrate that the models developed offer sufficiently accuratepredictions for practical designs while at the same time being quite straightforward to use

Research limitationsimplications ndash The analysis is based on a single microchannel while in apractical microchannel heat sink multiple channels are employed in parallel Therefore theoptimization should take into account the impact of inletoutlet headers Also a prescribed pumpingpower may be used as the design constraint instead of pressure head

Practical implications ndash Very useful design methodologies for practical design of microchannelheat sinks

Originalityvalue ndash Closed-form solutions from five analytical models are derived in a format thatcan be easily implemented in optimization procedures for minimizing the thermal resistance ofmicrochannel heat sinks

Keywords Optimization techniques Heat transfer Convection

Paper type Technical paper

The Emerald Research Register for this journal is available at The current issue and full text archive of this journal is available at

wwwemeraldinsightcomresearchregister wwwemeraldinsightcom0961-5539htm

NomenclatureAc frac14 microchannel cross-sectional areaAf frac14 fin cross-sectional areaAs frac14 area of heat sinkCp frac14 specific heatDh frac14 hydraulic diameterf Re frac14 friction constant

h frac14 heat transfer coefficientH frac14 height of heat sinkHc frac14 microchannel depthk frac14 thermal conductivityL frac14 length of heat sink_m frac14 mass flow rate

The authors acknowledge the financial support from members of the Cooling TechnologiesResearch Center (available at httpwidgetecnpurdueedu CTRC) a National ScienceFoundation IndustryUniversity Cooperative Research Center at Purdue University


7

Received February 2003Revised November 2003

Accepted May 2004

International Journal for NumericalMethods in Heat amp Fluid Flow

Vol 15 No 1 2005pp 7-26

q Emerald Group Publishing Limited0961-5539

DOI 10110809615530510571921








HFF151

8











Case1 2 3




resistances


9





2ksrsaquoT

rsaquoy

yfrac140


or

2ksdT

dy

yfrac140

frac14wc thorn ww

wwq00 eth6THORN



HFF151

10


q frac14 hwc

2L




2ksdT

dy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww

wwq00 eth8THORN





the base of the fin


11




Ro frac14DTmax

q00Aseth9THORN


Rcond frac14t
















teth12THORN

HFF151

12




Ro frac14DTmax


1




Rcond frac14t



Rconv frac141




Rcal frac141

rfOCpeth18THORN





13

d2T

dy 2frac14

2h



2ksdT

dy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww

wwq00 eth20THORN

dT

dy

yfrac14H c

frac14 0 eth21THORN

It follows that


m

q00

ks

2hfa

2hfathorn 1

wc thorn ww

ww





_mCpdT fethxTHORN







m

q00

ks

2hfa

2hfathorn 1

wc thorn ww

ww






R frac14DT


TethL 0THORN2 T0

q00ethLW THORN

frac141

m

1

ks

2hfa

2hfathorn 1

wc thorn ww

ww

coshethmH cTHORN

sinhethmH cTHORN

1

ethLW THORNthorn

ethwc thorn wwTHORN

rfCpumH cwc

1

Weth26THORN


rsaquo2T

rsaquoy 2frac14

2h


with

HFF151

14

2ksrsaquoT

rsaquoy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww


rsaquoT

rsaquoy

yfrac14H c

frac14 0 eth29THORN


rfCpumwcrsaquoT f



1

2rfCpumwcDh

rsaquoT f




Z X

0





Z X

0


eth33THORN

in which

b frac14a2

l 2eth34THORN

l 2 frac14kswwDh

2kfNueth35THORN




bs

s thorn 1

eth36THORN


2ksrsaquof

rsaquoY

Yfrac140

frac14Ja

seth37THORN

2ksrsaquof

rsaquoY

Yfrac14 ~Hc

frac14 0 eth38THORN



15


kssffiffiffig

pcosh

ffiffiffig

pethY 2 ~HcTHORN

sinh

ffiffiffig

p ~Hc

eth39THORN



frac14Ja

ks~Hcb


2ethY 2 ~HcTHORN

2 2b ~H

2

c

6

(

thorn 2X1nfrac141


sncos

npethY 2 ~HcTHORN

~Hc

esnX

)eth40THORN

in which


2

c

bthorn n 2p 2= ~H2

c


R frac14DT


TethL 0THORN2 T0


Ja

ks~Hcb


3b ~H

2

c

1






2ksrsaquoTs

rsaquozfrac14 2kf

rsaquoT f



T fethxTHORN frac14

Z wc=2

0


with

HFF151

16

um frac14

Z wc=2

0

v dz=wc=2


rsaquoT

rsaquoy

yfrac14H c

frac14rsaquoT

rsaquox

xfrac140

frac14rsaquoT

rsaquox

xfrac14L

frac14rsaquoT

rsaquoz

zfrac142ww

2

frac14rsaquoT f

rsaquoz

zfrac14wc

2

frac14 0 eth46THORN


ww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2

T thorn

rsaquoT

rsaquoz

0

2ww2

frac14 0 eth47THORN

in which

T frac14

Z 0

2ww2

T dz=ethww=2THORN


ksww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2



ksww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2




rsaquo

rsaquox

Z wc=2

0

uT f dz frac14kf

rfCp

rsaquoT f

rsaquoz

wc2

0

eth50THORN


umwc

2

rsaquo

rsaquoxT f thorn

1




DTceth52THORN

in which


17

DTc frac142hfa

2hfathorn 1


hH cq00 eth53THORN


A 2 rsaquo2

rsaquoX 2thorn

rsaquo2

rsaquoY 2

T 2 ethmH cTHORN


rsaquoT f


rsaquoT

rsaquoY

Yfrac140


rsaquoTs

rsaquoX

Xfrac141

frac14rsaquoTs

rsaquoY

Yfrac140

frac14rsaquoTs

rsaquoY

Yfrac141

frac14 0 eth57THORN



S frac14 hL

rfCpumwc

2







ifrac141

C0i

w0i


with

w01 frac14 2S

2thorn


2

2

thornmH c

A

2s

w02 frac14 2S

22


2

2

thornmH c

A

2s



HFF151

18



C03 frac14SA

mH c

2


R frac14DT


Teth1 0THORN2 T0

q00ethLW THORNDTc

frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


X2

ifrac141

C0i

w0i

ew0i thorn C03

1

ethLW THORN

eth61THORN


R frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


SA

mH c

2

1




2d

dxkplf thorn mf

d2

dy 2kulf 2

mf





U frac14 Dad2U

dY 22 P eth65THORN



U frac14kulf

um Da frac14

K

1H 2frac14

1

12a2s

Y frac14y

H P frac14

K

1mfum

dkplf

dx



19

U frac14 P cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 2 1

8gtltgt

9gt=gt eth67THORN




1rfCpkulf

rsaquokTlf


rsaquo2kTlf

rsaquoy 2eth69THORN



rsaquokTlrsaquoy

frac14rsaquokTlf





rsaquokTlf

rsaquoxfrac14

rsaquokTlrsaquox

frac14dTw


and


rsaquoxeth73THORN


d2u



dY 2eth75THORN

with


du

dYfrac14

duf


in which

HFF151

20

C frac141kf


haH 2


kTl2 Tw

q00Heth121THORNks

uf frac14kTlf 2 Tw

q00Heth121THORNks


uf frac14P

1 thorn C

242

1



Deth1 thorn CTHORN

CY

r 1A

2 C4 sinh


CY

r

thorn C5 cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt375

eth78THORN

u frac14 P

24Da cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt

21

2Y 2 thorn C1Y 2 Da

352 Cuf

eth79THORN

where



Da1 2 cosh

ffiffiffiffiffiffi1

Da

r ( )vuut

N 2 frac14C

Da


C

rsinh

ffiffiffiffiffiffi1

Da

r sinh


C

r

C1 frac14 1 2


pcosh

ffiffiffiffi1

Da

q 2 1

sinh

ffiffiffiffi1

Da

q


Deth1 thorn CTHORN

C3 frac14C



21


Deth1 thorn CTHORN


Ccosh


CY

q sinh

ffiffiffiffi1

Da

q D1

s

C5 frac14 Da 21

D1


R frac1412K

rfCpw3caW1 2um

2ufbH c



uf b frac14

Z 1

0

Uuf dY

13Z 1

0

U dy







HFF151

22

Tem

per

atu

reA

xia

lco

nd

uct

ion

Mod

elF

inF

luid

Fin

Flu

idG

over

nin

geq

uat

ion

sT

her

mal

resi

stan

ce(R

)

11D

1Dpound

poundN

otu

sed

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfQ

Cp

(88)

22D

1Dpound

pound

d2T

dy

22

hP

k sA

cethT

2T


0

_ mC

pdT

fethxTHORN

dx

frac14q

middotethw

cthorn

wwTHORN

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfC

pQ

(89)

32D

2Dpound

poundk srsaquo

2T

rsaquoy

2frac14

2h wwethT

2T

fethx

yTHORNTHORN

rfC

pu

mw

crsaquoT

f

rsaquox

frac142hethT

2T

fethx

yTHORNTHORN

Ja

k s~ H

cb

eth1thorn

L=aTHORNthorn

1 3b~ H

2 c

1

ethLW

THORN(9

0)

42D

2Dp

pound7

2TethxyzTHORNfrac14

0

7middotethk VT

fethx

yzTHORNTHORNfrac14

af7

2T

fethx

yzTHORN

2hfa

2hfa

thorn1

1

2nhH

ch

fLthorn

1

rfC

pQthorn

1

ethrfC

pQTHORN2

nk s

Hcw

w

L

(91)

51D

1Dpound

pound

2d dxkpl fthorn

mf

d2

dy

2kul f2

mf k1kul ffrac14

0

k sersaquo

2kT

lrsaquoy

2frac14

haethk

Tl2

kTl fTHORN

1r

fCpkul frsaquokT

l frsaquox

frac14haethk

Tl2

kTl fTHORNthorn

k fersaquo

2kT

l frsaquoy

2

12K

rfC

pw

3 ca

W1

2u

m2

uf

hH

c

eth12

1THORNk

sL

W(9

2)

Notespound

ndashn

otco

nsi

der

ed

andp

ndashco

nsi

der

ed


analytical models


23




f appRe frac1432

ethLthornTHORN057

2


1=2





Case




Case1 2 3

Nufd 597 581 606Nu 560 555 585


HFF151

24









rsaquoR


rsaquoR





1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072



25


References

















Further reading


HFF151

26








HFF151

8











Case1 2 3




resistances


9





2ksrsaquoT

rsaquoy

yfrac140


or

2ksdT

dy

yfrac140

frac14wc thorn ww

wwq00 eth6THORN



HFF151

10


q frac14 hwc

2L




2ksdT

dy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww

wwq00 eth8THORN





the base of the fin


11




Ro frac14DTmax

q00Aseth9THORN


Rcond frac14t
















teth12THORN

HFF151

12




Ro frac14DTmax


1




Rcond frac14t



Rconv frac141




Rcal frac141

rfOCpeth18THORN





13

d2T

dy 2frac14

2h



2ksdT

dy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww

wwq00 eth20THORN

dT

dy

yfrac14H c

frac14 0 eth21THORN

It follows that


m

q00

ks

2hfa

2hfathorn 1

wc thorn ww

ww





_mCpdT fethxTHORN







m

q00

ks

2hfa

2hfathorn 1

wc thorn ww

ww






R frac14DT


TethL 0THORN2 T0

q00ethLW THORN

frac141

m

1

ks

2hfa

2hfathorn 1

wc thorn ww

ww

coshethmH cTHORN

sinhethmH cTHORN

1

ethLW THORNthorn

ethwc thorn wwTHORN

rfCpumH cwc

1

Weth26THORN


rsaquo2T

rsaquoy 2frac14

2h


with

HFF151

14

2ksrsaquoT

rsaquoy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww


rsaquoT

rsaquoy

yfrac14H c

frac14 0 eth29THORN


rfCpumwcrsaquoT f



1

2rfCpumwcDh

rsaquoT f




Z X

0





Z X

0


eth33THORN

in which

b frac14a2

l 2eth34THORN

l 2 frac14kswwDh

2kfNueth35THORN




bs

s thorn 1

eth36THORN


2ksrsaquof

rsaquoY

Yfrac140

frac14Ja

seth37THORN

2ksrsaquof

rsaquoY

Yfrac14 ~Hc

frac14 0 eth38THORN



15


kssffiffiffig

pcosh

ffiffiffig

pethY 2 ~HcTHORN

sinh

ffiffiffig

p ~Hc

eth39THORN



frac14Ja

ks~Hcb


2ethY 2 ~HcTHORN

2 2b ~H

2

c

6

(

thorn 2X1nfrac141


sncos

npethY 2 ~HcTHORN

~Hc

esnX

)eth40THORN

in which


2

c

bthorn n 2p 2= ~H2

c


R frac14DT


TethL 0THORN2 T0


Ja

ks~Hcb


3b ~H

2

c

1






2ksrsaquoTs

rsaquozfrac14 2kf

rsaquoT f



T fethxTHORN frac14

Z wc=2

0


with

HFF151

16

um frac14

Z wc=2

0

v dz=wc=2


rsaquoT

rsaquoy

yfrac14H c

frac14rsaquoT

rsaquox

xfrac140

frac14rsaquoT

rsaquox

xfrac14L

frac14rsaquoT

rsaquoz

zfrac142ww

2

frac14rsaquoT f

rsaquoz

zfrac14wc

2

frac14 0 eth46THORN


ww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2

T thorn

rsaquoT

rsaquoz

0

2ww2

frac14 0 eth47THORN

in which

T frac14

Z 0

2ww2

T dz=ethww=2THORN


ksww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2



ksww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2




rsaquo

rsaquox

Z wc=2

0

uT f dz frac14kf

rfCp

rsaquoT f

rsaquoz

wc2

0

eth50THORN


umwc

2

rsaquo

rsaquoxT f thorn

1




DTceth52THORN

in which


17

DTc frac142hfa

2hfathorn 1


hH cq00 eth53THORN


A 2 rsaquo2

rsaquoX 2thorn

rsaquo2

rsaquoY 2

T 2 ethmH cTHORN


rsaquoT f


rsaquoT

rsaquoY

Yfrac140


rsaquoTs

rsaquoX

Xfrac141

frac14rsaquoTs

rsaquoY

Yfrac140

frac14rsaquoTs

rsaquoY

Yfrac141

frac14 0 eth57THORN



S frac14 hL

rfCpumwc

2







ifrac141

C0i

w0i


with

w01 frac14 2S

2thorn


2

2

thornmH c

A

2s

w02 frac14 2S

22


2

2

thornmH c

A

2s



HFF151

18



C03 frac14SA

mH c

2


R frac14DT


Teth1 0THORN2 T0

q00ethLW THORNDTc

frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


X2

ifrac141

C0i

w0i

ew0i thorn C03

1

ethLW THORN

eth61THORN


R frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


SA

mH c

2

1




2d

dxkplf thorn mf

d2

dy 2kulf 2

mf





U frac14 Dad2U

dY 22 P eth65THORN



U frac14kulf

um Da frac14

K

1H 2frac14

1

12a2s

Y frac14y

H P frac14

K

1mfum

dkplf

dx



19

U frac14 P cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 2 1

8gtltgt

9gt=gt eth67THORN




1rfCpkulf

rsaquokTlf


rsaquo2kTlf

rsaquoy 2eth69THORN



rsaquokTlrsaquoy

frac14rsaquokTlf





rsaquokTlf

rsaquoxfrac14

rsaquokTlrsaquox

frac14dTw


and


rsaquoxeth73THORN


d2u



dY 2eth75THORN

with


du

dYfrac14

duf


in which

HFF151

20

C frac141kf


haH 2


kTl2 Tw

q00Heth121THORNks

uf frac14kTlf 2 Tw

q00Heth121THORNks


uf frac14P

1 thorn C

242

1



Deth1 thorn CTHORN

CY

r 1A

2 C4 sinh


CY

r

thorn C5 cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt375

eth78THORN

u frac14 P

24Da cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt

21

2Y 2 thorn C1Y 2 Da

352 Cuf

eth79THORN

where



Da1 2 cosh

ffiffiffiffiffiffi1

Da

r ( )vuut

N 2 frac14C

Da


C

rsinh

ffiffiffiffiffiffi1

Da

r sinh


C

r

C1 frac14 1 2


pcosh

ffiffiffiffi1

Da

q 2 1

sinh

ffiffiffiffi1

Da

q


Deth1 thorn CTHORN

C3 frac14C



21


Deth1 thorn CTHORN


Ccosh


CY

q sinh

ffiffiffiffi1

Da

q D1

s

C5 frac14 Da 21

D1


R frac1412K

rfCpw3caW1 2um

2ufbH c



uf b frac14

Z 1

0

Uuf dY

13Z 1

0

U dy







HFF151

22

Tem

per

atu

reA

xia

lco

nd

uct

ion

Mod

elF

inF

luid

Fin

Flu

idG

over

nin

geq

uat

ion

sT

her

mal

resi

stan

ce(R

)

11D

1Dpound

poundN

otu

sed

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfQ

Cp

(88)

22D

1Dpound

pound

d2T

dy

22

hP

k sA

cethT

2T


0

_ mC

pdT

fethxTHORN

dx

frac14q

middotethw

cthorn

wwTHORN

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfC

pQ

(89)

32D

2Dpound

poundk srsaquo

2T

rsaquoy

2frac14

2h wwethT

2T

fethx

yTHORNTHORN

rfC

pu

mw

crsaquoT

f

rsaquox

frac142hethT

2T

fethx

yTHORNTHORN

Ja

k s~ H

cb

eth1thorn

L=aTHORNthorn

1 3b~ H

2 c

1

ethLW

THORN(9

0)

42D

2Dp

pound7

2TethxyzTHORNfrac14

0

7middotethk VT

fethx

yzTHORNTHORNfrac14

af7

2T

fethx

yzTHORN

2hfa

2hfa

thorn1

1

2nhH

ch

fLthorn

1

rfC

pQthorn

1

ethrfC

pQTHORN2

nk s

Hcw

w

L

(91)

51D

1Dpound

pound

2d dxkpl fthorn

mf

d2

dy

2kul f2

mf k1kul ffrac14

0

k sersaquo

2kT

lrsaquoy

2frac14

haethk

Tl2

kTl fTHORN

1r

fCpkul frsaquokT

l frsaquox

frac14haethk

Tl2

kTl fTHORNthorn

k fersaquo

2kT

l frsaquoy

2

12K

rfC

pw

3 ca

W1

2u

m2

uf

hH

c

eth12

1THORNk

sL

W(9

2)

Notespound

ndashn

otco

nsi

der

ed

andp

ndashco

nsi

der

ed


analytical models


23




f appRe frac1432

ethLthornTHORN057

2


1=2





Case




Case1 2 3

Nufd 597 581 606Nu 560 555 585


HFF151

24









rsaquoR


rsaquoR





1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072



25


References

















Further reading


HFF151

26











Case1 2 3




resistances


9





2ksrsaquoT

rsaquoy

yfrac140


or

2ksdT

dy

yfrac140

frac14wc thorn ww

wwq00 eth6THORN



HFF151

10


q frac14 hwc

2L




2ksdT

dy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww

wwq00 eth8THORN





the base of the fin


11




Ro frac14DTmax

q00Aseth9THORN


Rcond frac14t
















teth12THORN

HFF151

12




Ro frac14DTmax


1




Rcond frac14t



Rconv frac141




Rcal frac141

rfOCpeth18THORN





13

d2T

dy 2frac14

2h



2ksdT

dy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww

wwq00 eth20THORN

dT

dy

yfrac14H c

frac14 0 eth21THORN

It follows that


m

q00

ks

2hfa

2hfathorn 1

wc thorn ww

ww





_mCpdT fethxTHORN







m

q00

ks

2hfa

2hfathorn 1

wc thorn ww

ww






R frac14DT


TethL 0THORN2 T0

q00ethLW THORN

frac141

m

1

ks

2hfa

2hfathorn 1

wc thorn ww

ww

coshethmH cTHORN

sinhethmH cTHORN

1

ethLW THORNthorn

ethwc thorn wwTHORN

rfCpumH cwc

1

Weth26THORN


rsaquo2T

rsaquoy 2frac14

2h


with

HFF151

14

2ksrsaquoT

rsaquoy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww


rsaquoT

rsaquoy

yfrac14H c

frac14 0 eth29THORN


rfCpumwcrsaquoT f



1

2rfCpumwcDh

rsaquoT f




Z X

0





Z X

0


eth33THORN

in which

b frac14a2

l 2eth34THORN

l 2 frac14kswwDh

2kfNueth35THORN




bs

s thorn 1

eth36THORN


2ksrsaquof

rsaquoY

Yfrac140

frac14Ja

seth37THORN

2ksrsaquof

rsaquoY

Yfrac14 ~Hc

frac14 0 eth38THORN



15


kssffiffiffig

pcosh

ffiffiffig

pethY 2 ~HcTHORN

sinh

ffiffiffig

p ~Hc

eth39THORN



frac14Ja

ks~Hcb


2ethY 2 ~HcTHORN

2 2b ~H

2

c

6

(

thorn 2X1nfrac141


sncos

npethY 2 ~HcTHORN

~Hc

esnX

)eth40THORN

in which


2

c

bthorn n 2p 2= ~H2

c


R frac14DT


TethL 0THORN2 T0


Ja

ks~Hcb


3b ~H

2

c

1






2ksrsaquoTs

rsaquozfrac14 2kf

rsaquoT f



T fethxTHORN frac14

Z wc=2

0


with

HFF151

16

um frac14

Z wc=2

0

v dz=wc=2


rsaquoT

rsaquoy

yfrac14H c

frac14rsaquoT

rsaquox

xfrac140

frac14rsaquoT

rsaquox

xfrac14L

frac14rsaquoT

rsaquoz

zfrac142ww

2

frac14rsaquoT f

rsaquoz

zfrac14wc

2

frac14 0 eth46THORN


ww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2

T thorn

rsaquoT

rsaquoz

0

2ww2

frac14 0 eth47THORN

in which

T frac14

Z 0

2ww2

T dz=ethww=2THORN


ksww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2



ksww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2




rsaquo

rsaquox

Z wc=2

0

uT f dz frac14kf

rfCp

rsaquoT f

rsaquoz

wc2

0

eth50THORN


umwc

2

rsaquo

rsaquoxT f thorn

1




DTceth52THORN

in which


17

DTc frac142hfa

2hfathorn 1


hH cq00 eth53THORN


A 2 rsaquo2

rsaquoX 2thorn

rsaquo2

rsaquoY 2

T 2 ethmH cTHORN


rsaquoT f


rsaquoT

rsaquoY

Yfrac140


rsaquoTs

rsaquoX

Xfrac141

frac14rsaquoTs

rsaquoY

Yfrac140

frac14rsaquoTs

rsaquoY

Yfrac141

frac14 0 eth57THORN



S frac14 hL

rfCpumwc

2







ifrac141

C0i

w0i


with

w01 frac14 2S

2thorn


2

2

thornmH c

A

2s

w02 frac14 2S

22


2

2

thornmH c

A

2s



HFF151

18



C03 frac14SA

mH c

2


R frac14DT


Teth1 0THORN2 T0

q00ethLW THORNDTc

frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


X2

ifrac141

C0i

w0i

ew0i thorn C03

1

ethLW THORN

eth61THORN


R frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


SA

mH c

2

1




2d

dxkplf thorn mf

d2

dy 2kulf 2

mf





U frac14 Dad2U

dY 22 P eth65THORN



U frac14kulf

um Da frac14

K

1H 2frac14

1

12a2s

Y frac14y

H P frac14

K

1mfum

dkplf

dx



19

U frac14 P cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 2 1

8gtltgt

9gt=gt eth67THORN




1rfCpkulf

rsaquokTlf


rsaquo2kTlf

rsaquoy 2eth69THORN



rsaquokTlrsaquoy

frac14rsaquokTlf





rsaquokTlf

rsaquoxfrac14

rsaquokTlrsaquox

frac14dTw


and


rsaquoxeth73THORN


d2u



dY 2eth75THORN

with


du

dYfrac14

duf


in which

HFF151

20

C frac141kf


haH 2


kTl2 Tw

q00Heth121THORNks

uf frac14kTlf 2 Tw

q00Heth121THORNks


uf frac14P

1 thorn C

242

1



Deth1 thorn CTHORN

CY

r 1A

2 C4 sinh


CY

r

thorn C5 cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt375

eth78THORN

u frac14 P

24Da cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt

21

2Y 2 thorn C1Y 2 Da

352 Cuf

eth79THORN

where



Da1 2 cosh

ffiffiffiffiffiffi1

Da

r ( )vuut

N 2 frac14C

Da


C

rsinh

ffiffiffiffiffiffi1

Da

r sinh


C

r

C1 frac14 1 2


pcosh

ffiffiffiffi1

Da

q 2 1

sinh

ffiffiffiffi1

Da

q


Deth1 thorn CTHORN

C3 frac14C



21


Deth1 thorn CTHORN


Ccosh


CY

q sinh

ffiffiffiffi1

Da

q D1

s

C5 frac14 Da 21

D1


R frac1412K

rfCpw3caW1 2um

2ufbH c



uf b frac14

Z 1

0

Uuf dY

13Z 1

0

U dy







HFF151

22

Tem

per

atu

reA

xia

lco

nd

uct

ion

Mod

elF

inF

luid

Fin

Flu

idG

over

nin

geq

uat

ion

sT

her

mal

resi

stan

ce(R

)

11D

1Dpound

poundN

otu

sed

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfQ

Cp

(88)

22D

1Dpound

pound

d2T

dy

22

hP

k sA

cethT

2T


0

_ mC

pdT

fethxTHORN

dx

frac14q

middotethw

cthorn

wwTHORN

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfC

pQ

(89)

32D

2Dpound

poundk srsaquo

2T

rsaquoy

2frac14

2h wwethT

2T

fethx

yTHORNTHORN

rfC

pu

mw

crsaquoT

f

rsaquox

frac142hethT

2T

fethx

yTHORNTHORN

Ja

k s~ H

cb

eth1thorn

L=aTHORNthorn

1 3b~ H

2 c

1

ethLW

THORN(9

0)

42D

2Dp

pound7

2TethxyzTHORNfrac14

0

7middotethk VT

fethx

yzTHORNTHORNfrac14

af7

2T

fethx

yzTHORN

2hfa

2hfa

thorn1

1

2nhH

ch

fLthorn

1

rfC

pQthorn

1

ethrfC

pQTHORN2

nk s

Hcw

w

L

(91)

51D

1Dpound

pound

2d dxkpl fthorn

mf

d2

dy

2kul f2

mf k1kul ffrac14

0

k sersaquo

2kT

lrsaquoy

2frac14

haethk

Tl2

kTl fTHORN

1r

fCpkul frsaquokT

l frsaquox

frac14haethk

Tl2

kTl fTHORNthorn

k fersaquo

2kT

l frsaquoy

2

12K

rfC

pw

3 ca

W1

2u

m2

uf

hH

c

eth12

1THORNk

sL

W(9

2)

Notespound

ndashn

otco

nsi

der

ed

andp

ndashco

nsi

der

ed


analytical models


23




f appRe frac1432

ethLthornTHORN057

2


1=2





Case




Case1 2 3

Nufd 597 581 606Nu 560 555 585


HFF151

24









rsaquoR


rsaquoR





1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072



25


References

















Further reading


HFF151

26





2ksrsaquoT

rsaquoy

yfrac140


or

2ksdT

dy

yfrac140

frac14wc thorn ww

wwq00 eth6THORN



HFF151

10


q frac14 hwc

2L




2ksdT

dy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww

wwq00 eth8THORN





the base of the fin


11




Ro frac14DTmax

q00Aseth9THORN


Rcond frac14t
















teth12THORN

HFF151

12




Ro frac14DTmax


1




Rcond frac14t



Rconv frac141




Rcal frac141

rfOCpeth18THORN





13

d2T

dy 2frac14

2h



2ksdT

dy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww

wwq00 eth20THORN

dT

dy

yfrac14H c

frac14 0 eth21THORN

It follows that


m

q00

ks

2hfa

2hfathorn 1

wc thorn ww

ww





_mCpdT fethxTHORN







m

q00

ks

2hfa

2hfathorn 1

wc thorn ww

ww






R frac14DT


TethL 0THORN2 T0

q00ethLW THORN

frac141

m

1

ks

2hfa

2hfathorn 1

wc thorn ww

ww

coshethmH cTHORN

sinhethmH cTHORN

1

ethLW THORNthorn

ethwc thorn wwTHORN

rfCpumH cwc

1

Weth26THORN


rsaquo2T

rsaquoy 2frac14

2h


with

HFF151

14

2ksrsaquoT

rsaquoy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww


rsaquoT

rsaquoy

yfrac14H c

frac14 0 eth29THORN


rfCpumwcrsaquoT f



1

2rfCpumwcDh

rsaquoT f




Z X

0





Z X

0


eth33THORN

in which

b frac14a2

l 2eth34THORN

l 2 frac14kswwDh

2kfNueth35THORN




bs

s thorn 1

eth36THORN


2ksrsaquof

rsaquoY

Yfrac140

frac14Ja

seth37THORN

2ksrsaquof

rsaquoY

Yfrac14 ~Hc

frac14 0 eth38THORN



15


kssffiffiffig

pcosh

ffiffiffig

pethY 2 ~HcTHORN

sinh

ffiffiffig

p ~Hc

eth39THORN



frac14Ja

ks~Hcb


2ethY 2 ~HcTHORN

2 2b ~H

2

c

6

(

thorn 2X1nfrac141


sncos

npethY 2 ~HcTHORN

~Hc

esnX

)eth40THORN

in which


2

c

bthorn n 2p 2= ~H2

c


R frac14DT


TethL 0THORN2 T0


Ja

ks~Hcb


3b ~H

2

c

1






2ksrsaquoTs

rsaquozfrac14 2kf

rsaquoT f



T fethxTHORN frac14

Z wc=2

0


with

HFF151

16

um frac14

Z wc=2

0

v dz=wc=2


rsaquoT

rsaquoy

yfrac14H c

frac14rsaquoT

rsaquox

xfrac140

frac14rsaquoT

rsaquox

xfrac14L

frac14rsaquoT

rsaquoz

zfrac142ww

2

frac14rsaquoT f

rsaquoz

zfrac14wc

2

frac14 0 eth46THORN


ww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2

T thorn

rsaquoT

rsaquoz

0

2ww2

frac14 0 eth47THORN

in which

T frac14

Z 0

2ww2

T dz=ethww=2THORN


ksww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2



ksww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2




rsaquo

rsaquox

Z wc=2

0

uT f dz frac14kf

rfCp

rsaquoT f

rsaquoz

wc2

0

eth50THORN


umwc

2

rsaquo

rsaquoxT f thorn

1




DTceth52THORN

in which


17

DTc frac142hfa

2hfathorn 1


hH cq00 eth53THORN


A 2 rsaquo2

rsaquoX 2thorn

rsaquo2

rsaquoY 2

T 2 ethmH cTHORN


rsaquoT f


rsaquoT

rsaquoY

Yfrac140


rsaquoTs

rsaquoX

Xfrac141

frac14rsaquoTs

rsaquoY

Yfrac140

frac14rsaquoTs

rsaquoY

Yfrac141

frac14 0 eth57THORN



S frac14 hL

rfCpumwc

2







ifrac141

C0i

w0i


with

w01 frac14 2S

2thorn


2

2

thornmH c

A

2s

w02 frac14 2S

22


2

2

thornmH c

A

2s



HFF151

18



C03 frac14SA

mH c

2


R frac14DT


Teth1 0THORN2 T0

q00ethLW THORNDTc

frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


X2

ifrac141

C0i

w0i

ew0i thorn C03

1

ethLW THORN

eth61THORN


R frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


SA

mH c

2

1




2d

dxkplf thorn mf

d2

dy 2kulf 2

mf





U frac14 Dad2U

dY 22 P eth65THORN



U frac14kulf

um Da frac14

K

1H 2frac14

1

12a2s

Y frac14y

H P frac14

K

1mfum

dkplf

dx



19

U frac14 P cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 2 1

8gtltgt

9gt=gt eth67THORN




1rfCpkulf

rsaquokTlf


rsaquo2kTlf

rsaquoy 2eth69THORN



rsaquokTlrsaquoy

frac14rsaquokTlf





rsaquokTlf

rsaquoxfrac14

rsaquokTlrsaquox

frac14dTw


and


rsaquoxeth73THORN


d2u



dY 2eth75THORN

with


du

dYfrac14

duf


in which

HFF151

20

C frac141kf


haH 2


kTl2 Tw

q00Heth121THORNks

uf frac14kTlf 2 Tw

q00Heth121THORNks


uf frac14P

1 thorn C

242

1



Deth1 thorn CTHORN

CY

r 1A

2 C4 sinh


CY

r

thorn C5 cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt375

eth78THORN

u frac14 P

24Da cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt

21

2Y 2 thorn C1Y 2 Da

352 Cuf

eth79THORN

where



Da1 2 cosh

ffiffiffiffiffiffi1

Da

r ( )vuut

N 2 frac14C

Da


C

rsinh

ffiffiffiffiffiffi1

Da

r sinh


C

r

C1 frac14 1 2


pcosh

ffiffiffiffi1

Da

q 2 1

sinh

ffiffiffiffi1

Da

q


Deth1 thorn CTHORN

C3 frac14C



21


Deth1 thorn CTHORN


Ccosh


CY

q sinh

ffiffiffiffi1

Da

q D1

s

C5 frac14 Da 21

D1


R frac1412K

rfCpw3caW1 2um

2ufbH c



uf b frac14

Z 1

0

Uuf dY

13Z 1

0

U dy







HFF151

22

Tem

per

atu

reA

xia

lco

nd

uct

ion

Mod

elF

inF

luid

Fin

Flu

idG

over

nin

geq

uat

ion

sT

her

mal

resi

stan

ce(R

)

11D

1Dpound

poundN

otu

sed

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfQ

Cp

(88)

22D

1Dpound

pound

d2T

dy

22

hP

k sA

cethT

2T


0

_ mC

pdT

fethxTHORN

dx

frac14q

middotethw

cthorn

wwTHORN

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfC

pQ

(89)

32D

2Dpound

poundk srsaquo

2T

rsaquoy

2frac14

2h wwethT

2T

fethx

yTHORNTHORN

rfC

pu

mw

crsaquoT

f

rsaquox

frac142hethT

2T

fethx

yTHORNTHORN

Ja

k s~ H

cb

eth1thorn

L=aTHORNthorn

1 3b~ H

2 c

1

ethLW

THORN(9

0)

42D

2Dp

pound7

2TethxyzTHORNfrac14

0

7middotethk VT

fethx

yzTHORNTHORNfrac14

af7

2T

fethx

yzTHORN

2hfa

2hfa

thorn1

1

2nhH

ch

fLthorn

1

rfC

pQthorn

1

ethrfC

pQTHORN2

nk s

Hcw

w

L

(91)

51D

1Dpound

pound

2d dxkpl fthorn

mf

d2

dy

2kul f2

mf k1kul ffrac14

0

k sersaquo

2kT

lrsaquoy

2frac14

haethk

Tl2

kTl fTHORN

1r

fCpkul frsaquokT

l frsaquox

frac14haethk

Tl2

kTl fTHORNthorn

k fersaquo

2kT

l frsaquoy

2

12K

rfC

pw

3 ca

W1

2u

m2

uf

hH

c

eth12

1THORNk

sL

W(9

2)

Notespound

ndashn

otco

nsi

der

ed

andp

ndashco

nsi

der

ed


analytical models


23




f appRe frac1432

ethLthornTHORN057

2


1=2





Case




Case1 2 3

Nufd 597 581 606Nu 560 555 585


HFF151

24









rsaquoR


rsaquoR





1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072



25


References

















Further reading


HFF151

26


q frac14 hwc

2L




2ksdT

dy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww

wwq00 eth8THORN





the base of the fin


11




Ro frac14DTmax

q00Aseth9THORN


Rcond frac14t
















teth12THORN

HFF151

12




Ro frac14DTmax


1




Rcond frac14t



Rconv frac141




Rcal frac141

rfOCpeth18THORN





13

d2T

dy 2frac14

2h



2ksdT

dy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww

wwq00 eth20THORN

dT

dy

yfrac14H c

frac14 0 eth21THORN

It follows that


m

q00

ks

2hfa

2hfathorn 1

wc thorn ww

ww





_mCpdT fethxTHORN







m

q00

ks

2hfa

2hfathorn 1

wc thorn ww

ww






R frac14DT


TethL 0THORN2 T0

q00ethLW THORN

frac141

m

1

ks

2hfa

2hfathorn 1

wc thorn ww

ww

coshethmH cTHORN

sinhethmH cTHORN

1

ethLW THORNthorn

ethwc thorn wwTHORN

rfCpumH cwc

1

Weth26THORN


rsaquo2T

rsaquoy 2frac14

2h


with

HFF151

14

2ksrsaquoT

rsaquoy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww


rsaquoT

rsaquoy

yfrac14H c

frac14 0 eth29THORN


rfCpumwcrsaquoT f



1

2rfCpumwcDh

rsaquoT f




Z X

0





Z X

0


eth33THORN

in which

b frac14a2

l 2eth34THORN

l 2 frac14kswwDh

2kfNueth35THORN




bs

s thorn 1

eth36THORN


2ksrsaquof

rsaquoY

Yfrac140

frac14Ja

seth37THORN

2ksrsaquof

rsaquoY

Yfrac14 ~Hc

frac14 0 eth38THORN



15


kssffiffiffig

pcosh

ffiffiffig

pethY 2 ~HcTHORN

sinh

ffiffiffig

p ~Hc

eth39THORN



frac14Ja

ks~Hcb


2ethY 2 ~HcTHORN

2 2b ~H

2

c

6

(

thorn 2X1nfrac141


sncos

npethY 2 ~HcTHORN

~Hc

esnX

)eth40THORN

in which


2

c

bthorn n 2p 2= ~H2

c


R frac14DT


TethL 0THORN2 T0


Ja

ks~Hcb


3b ~H

2

c

1






2ksrsaquoTs

rsaquozfrac14 2kf

rsaquoT f



T fethxTHORN frac14

Z wc=2

0


with

HFF151

16

um frac14

Z wc=2

0

v dz=wc=2


rsaquoT

rsaquoy

yfrac14H c

frac14rsaquoT

rsaquox

xfrac140

frac14rsaquoT

rsaquox

xfrac14L

frac14rsaquoT

rsaquoz

zfrac142ww

2

frac14rsaquoT f

rsaquoz

zfrac14wc

2

frac14 0 eth46THORN


ww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2

T thorn

rsaquoT

rsaquoz

0

2ww2

frac14 0 eth47THORN

in which

T frac14

Z 0

2ww2

T dz=ethww=2THORN


ksww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2



ksww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2




rsaquo

rsaquox

Z wc=2

0

uT f dz frac14kf

rfCp

rsaquoT f

rsaquoz

wc2

0

eth50THORN


umwc

2

rsaquo

rsaquoxT f thorn

1




DTceth52THORN

in which


17

DTc frac142hfa

2hfathorn 1


hH cq00 eth53THORN


A 2 rsaquo2

rsaquoX 2thorn

rsaquo2

rsaquoY 2

T 2 ethmH cTHORN


rsaquoT f


rsaquoT

rsaquoY

Yfrac140


rsaquoTs

rsaquoX

Xfrac141

frac14rsaquoTs

rsaquoY

Yfrac140

frac14rsaquoTs

rsaquoY

Yfrac141

frac14 0 eth57THORN



S frac14 hL

rfCpumwc

2







ifrac141

C0i

w0i


with

w01 frac14 2S

2thorn


2

2

thornmH c

A

2s

w02 frac14 2S

22


2

2

thornmH c

A

2s



HFF151

18



C03 frac14SA

mH c

2


R frac14DT


Teth1 0THORN2 T0

q00ethLW THORNDTc

frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


X2

ifrac141

C0i

w0i

ew0i thorn C03

1

ethLW THORN

eth61THORN


R frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


SA

mH c

2

1




2d

dxkplf thorn mf

d2

dy 2kulf 2

mf





U frac14 Dad2U

dY 22 P eth65THORN



U frac14kulf

um Da frac14

K

1H 2frac14

1

12a2s

Y frac14y

H P frac14

K

1mfum

dkplf

dx



19

U frac14 P cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 2 1

8gtltgt

9gt=gt eth67THORN




1rfCpkulf

rsaquokTlf


rsaquo2kTlf

rsaquoy 2eth69THORN



rsaquokTlrsaquoy

frac14rsaquokTlf





rsaquokTlf

rsaquoxfrac14

rsaquokTlrsaquox

frac14dTw


and


rsaquoxeth73THORN


d2u



dY 2eth75THORN

with


du

dYfrac14

duf


in which

HFF151

20

C frac141kf


haH 2


kTl2 Tw

q00Heth121THORNks

uf frac14kTlf 2 Tw

q00Heth121THORNks


uf frac14P

1 thorn C

242

1



Deth1 thorn CTHORN

CY

r 1A

2 C4 sinh


CY

r

thorn C5 cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt375

eth78THORN

u frac14 P

24Da cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt

21

2Y 2 thorn C1Y 2 Da

352 Cuf

eth79THORN

where



Da1 2 cosh

ffiffiffiffiffiffi1

Da

r ( )vuut

N 2 frac14C

Da


C

rsinh

ffiffiffiffiffiffi1

Da

r sinh


C

r

C1 frac14 1 2


pcosh

ffiffiffiffi1

Da

q 2 1

sinh

ffiffiffiffi1

Da

q


Deth1 thorn CTHORN

C3 frac14C



21


Deth1 thorn CTHORN


Ccosh


CY

q sinh

ffiffiffiffi1

Da

q D1

s

C5 frac14 Da 21

D1


R frac1412K

rfCpw3caW1 2um

2ufbH c



uf b frac14

Z 1

0

Uuf dY

13Z 1

0

U dy







HFF151

22

Tem

per

atu

reA

xia

lco

nd

uct

ion

Mod

elF

inF

luid

Fin

Flu

idG

over

nin

geq

uat

ion

sT

her

mal

resi

stan

ce(R

)

11D

1Dpound

poundN

otu

sed

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfQ

Cp

(88)

22D

1Dpound

pound

d2T

dy

22

hP

k sA

cethT

2T


0

_ mC

pdT

fethxTHORN

dx

frac14q

middotethw

cthorn

wwTHORN

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfC

pQ

(89)

32D

2Dpound

poundk srsaquo

2T

rsaquoy

2frac14

2h wwethT

2T

fethx

yTHORNTHORN

rfC

pu

mw

crsaquoT

f

rsaquox

frac142hethT

2T

fethx

yTHORNTHORN

Ja

k s~ H

cb

eth1thorn

L=aTHORNthorn

1 3b~ H

2 c

1

ethLW

THORN(9

0)

42D

2Dp

pound7

2TethxyzTHORNfrac14

0

7middotethk VT

fethx

yzTHORNTHORNfrac14

af7

2T

fethx

yzTHORN

2hfa

2hfa

thorn1

1

2nhH

ch

fLthorn

1

rfC

pQthorn

1

ethrfC

pQTHORN2

nk s

Hcw

w

L

(91)

51D

1Dpound

pound

2d dxkpl fthorn

mf

d2

dy

2kul f2

mf k1kul ffrac14

0

k sersaquo

2kT

lrsaquoy

2frac14

haethk

Tl2

kTl fTHORN

1r

fCpkul frsaquokT

l frsaquox

frac14haethk

Tl2

kTl fTHORNthorn

k fersaquo

2kT

l frsaquoy

2

12K

rfC

pw

3 ca

W1

2u

m2

uf

hH

c

eth12

1THORNk

sL

W(9

2)

Notespound

ndashn

otco

nsi

der

ed

andp

ndashco

nsi

der

ed


analytical models


23




f appRe frac1432

ethLthornTHORN057

2


1=2





Case




Case1 2 3

Nufd 597 581 606Nu 560 555 585


HFF151

24









rsaquoR


rsaquoR





1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072



25


References

















Further reading


HFF151

26




Ro frac14DTmax

q00Aseth9THORN


Rcond frac14t
















teth12THORN

HFF151

12




Ro frac14DTmax


1




Rcond frac14t



Rconv frac141




Rcal frac141

rfOCpeth18THORN





13

d2T

dy 2frac14

2h



2ksdT

dy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww

wwq00 eth20THORN

dT

dy

yfrac14H c

frac14 0 eth21THORN

It follows that


m

q00

ks

2hfa

2hfathorn 1

wc thorn ww

ww





_mCpdT fethxTHORN







m

q00

ks

2hfa

2hfathorn 1

wc thorn ww

ww






R frac14DT


TethL 0THORN2 T0

q00ethLW THORN

frac141

m

1

ks

2hfa

2hfathorn 1

wc thorn ww

ww

coshethmH cTHORN

sinhethmH cTHORN

1

ethLW THORNthorn

ethwc thorn wwTHORN

rfCpumH cwc

1

Weth26THORN


rsaquo2T

rsaquoy 2frac14

2h


with

HFF151

14

2ksrsaquoT

rsaquoy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww


rsaquoT

rsaquoy

yfrac14H c

frac14 0 eth29THORN


rfCpumwcrsaquoT f



1

2rfCpumwcDh

rsaquoT f




Z X

0





Z X

0


eth33THORN

in which

b frac14a2

l 2eth34THORN

l 2 frac14kswwDh

2kfNueth35THORN




bs

s thorn 1

eth36THORN


2ksrsaquof

rsaquoY

Yfrac140

frac14Ja

seth37THORN

2ksrsaquof

rsaquoY

Yfrac14 ~Hc

frac14 0 eth38THORN



15


kssffiffiffig

pcosh

ffiffiffig

pethY 2 ~HcTHORN

sinh

ffiffiffig

p ~Hc

eth39THORN



frac14Ja

ks~Hcb


2ethY 2 ~HcTHORN

2 2b ~H

2

c

6

(

thorn 2X1nfrac141


sncos

npethY 2 ~HcTHORN

~Hc

esnX

)eth40THORN

in which


2

c

bthorn n 2p 2= ~H2

c


R frac14DT


TethL 0THORN2 T0


Ja

ks~Hcb


3b ~H

2

c

1






2ksrsaquoTs

rsaquozfrac14 2kf

rsaquoT f



T fethxTHORN frac14

Z wc=2

0


with

HFF151

16

um frac14

Z wc=2

0

v dz=wc=2


rsaquoT

rsaquoy

yfrac14H c

frac14rsaquoT

rsaquox

xfrac140

frac14rsaquoT

rsaquox

xfrac14L

frac14rsaquoT

rsaquoz

zfrac142ww

2

frac14rsaquoT f

rsaquoz

zfrac14wc

2

frac14 0 eth46THORN


ww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2

T thorn

rsaquoT

rsaquoz

0

2ww2

frac14 0 eth47THORN

in which

T frac14

Z 0

2ww2

T dz=ethww=2THORN


ksww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2



ksww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2




rsaquo

rsaquox

Z wc=2

0

uT f dz frac14kf

rfCp

rsaquoT f

rsaquoz

wc2

0

eth50THORN


umwc

2

rsaquo

rsaquoxT f thorn

1




DTceth52THORN

in which


17

DTc frac142hfa

2hfathorn 1


hH cq00 eth53THORN


A 2 rsaquo2

rsaquoX 2thorn

rsaquo2

rsaquoY 2

T 2 ethmH cTHORN


rsaquoT f


rsaquoT

rsaquoY

Yfrac140


rsaquoTs

rsaquoX

Xfrac141

frac14rsaquoTs

rsaquoY

Yfrac140

frac14rsaquoTs

rsaquoY

Yfrac141

frac14 0 eth57THORN



S frac14 hL

rfCpumwc

2







ifrac141

C0i

w0i


with

w01 frac14 2S

2thorn


2

2

thornmH c

A

2s

w02 frac14 2S

22


2

2

thornmH c

A

2s



HFF151

18



C03 frac14SA

mH c

2


R frac14DT


Teth1 0THORN2 T0

q00ethLW THORNDTc

frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


X2

ifrac141

C0i

w0i

ew0i thorn C03

1

ethLW THORN

eth61THORN


R frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


SA

mH c

2

1




2d

dxkplf thorn mf

d2

dy 2kulf 2

mf





U frac14 Dad2U

dY 22 P eth65THORN



U frac14kulf

um Da frac14

K

1H 2frac14

1

12a2s

Y frac14y

H P frac14

K

1mfum

dkplf

dx



19

U frac14 P cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 2 1

8gtltgt

9gt=gt eth67THORN




1rfCpkulf

rsaquokTlf


rsaquo2kTlf

rsaquoy 2eth69THORN



rsaquokTlrsaquoy

frac14rsaquokTlf





rsaquokTlf

rsaquoxfrac14

rsaquokTlrsaquox

frac14dTw


and


rsaquoxeth73THORN


d2u



dY 2eth75THORN

with


du

dYfrac14

duf


in which

HFF151

20

C frac141kf


haH 2


kTl2 Tw

q00Heth121THORNks

uf frac14kTlf 2 Tw

q00Heth121THORNks


uf frac14P

1 thorn C

242

1



Deth1 thorn CTHORN

CY

r 1A

2 C4 sinh


CY

r

thorn C5 cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt375

eth78THORN

u frac14 P

24Da cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt

21

2Y 2 thorn C1Y 2 Da

352 Cuf

eth79THORN

where



Da1 2 cosh

ffiffiffiffiffiffi1

Da

r ( )vuut

N 2 frac14C

Da


C

rsinh

ffiffiffiffiffiffi1

Da

r sinh


C

r

C1 frac14 1 2


pcosh

ffiffiffiffi1

Da

q 2 1

sinh

ffiffiffiffi1

Da

q


Deth1 thorn CTHORN

C3 frac14C



21


Deth1 thorn CTHORN


Ccosh


CY

q sinh

ffiffiffiffi1

Da

q D1

s

C5 frac14 Da 21

D1


R frac1412K

rfCpw3caW1 2um

2ufbH c



uf b frac14

Z 1

0

Uuf dY

13Z 1

0

U dy







HFF151

22

Tem

per

atu

reA

xia

lco

nd

uct

ion

Mod

elF

inF

luid

Fin

Flu

idG

over

nin

geq

uat

ion

sT

her

mal

resi

stan

ce(R

)

11D

1Dpound

poundN

otu

sed

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfQ

Cp

(88)

22D

1Dpound

pound

d2T

dy

22

hP

k sA

cethT

2T


0

_ mC

pdT

fethxTHORN

dx

frac14q

middotethw

cthorn

wwTHORN

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfC

pQ

(89)

32D

2Dpound

poundk srsaquo

2T

rsaquoy

2frac14

2h wwethT

2T

fethx

yTHORNTHORN

rfC

pu

mw

crsaquoT

f

rsaquox

frac142hethT

2T

fethx

yTHORNTHORN

Ja

k s~ H

cb

eth1thorn

L=aTHORNthorn

1 3b~ H

2 c

1

ethLW

THORN(9

0)

42D

2Dp

pound7

2TethxyzTHORNfrac14

0

7middotethk VT

fethx

yzTHORNTHORNfrac14

af7

2T

fethx

yzTHORN

2hfa

2hfa

thorn1

1

2nhH

ch

fLthorn

1

rfC

pQthorn

1

ethrfC

pQTHORN2

nk s

Hcw

w

L

(91)

51D

1Dpound

pound

2d dxkpl fthorn

mf

d2

dy

2kul f2

mf k1kul ffrac14

0

k sersaquo

2kT

lrsaquoy

2frac14

haethk

Tl2

kTl fTHORN

1r

fCpkul frsaquokT

l frsaquox

frac14haethk

Tl2

kTl fTHORNthorn

k fersaquo

2kT

l frsaquoy

2

12K

rfC

pw

3 ca

W1

2u

m2

uf

hH

c

eth12

1THORNk

sL

W(9

2)

Notespound

ndashn

otco

nsi

der

ed

andp

ndashco

nsi

der

ed


analytical models


23




f appRe frac1432

ethLthornTHORN057

2


1=2





Case




Case1 2 3

Nufd 597 581 606Nu 560 555 585


HFF151

24









rsaquoR


rsaquoR





1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072



25


References

















Further reading


HFF151

26




Ro frac14DTmax


1




Rcond frac14t



Rconv frac141




Rcal frac141

rfOCpeth18THORN





13

d2T

dy 2frac14

2h



2ksdT

dy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww

wwq00 eth20THORN

dT

dy

yfrac14H c

frac14 0 eth21THORN

It follows that


m

q00

ks

2hfa

2hfathorn 1

wc thorn ww

ww





_mCpdT fethxTHORN







m

q00

ks

2hfa

2hfathorn 1

wc thorn ww

ww






R frac14DT


TethL 0THORN2 T0

q00ethLW THORN

frac141

m

1

ks

2hfa

2hfathorn 1

wc thorn ww

ww

coshethmH cTHORN

sinhethmH cTHORN

1

ethLW THORNthorn

ethwc thorn wwTHORN

rfCpumH cwc

1

Weth26THORN


rsaquo2T

rsaquoy 2frac14

2h


with

HFF151

14

2ksrsaquoT

rsaquoy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww


rsaquoT

rsaquoy

yfrac14H c

frac14 0 eth29THORN


rfCpumwcrsaquoT f



1

2rfCpumwcDh

rsaquoT f




Z X

0





Z X

0


eth33THORN

in which

b frac14a2

l 2eth34THORN

l 2 frac14kswwDh

2kfNueth35THORN




bs

s thorn 1

eth36THORN


2ksrsaquof

rsaquoY

Yfrac140

frac14Ja

seth37THORN

2ksrsaquof

rsaquoY

Yfrac14 ~Hc

frac14 0 eth38THORN



15


kssffiffiffig

pcosh

ffiffiffig

pethY 2 ~HcTHORN

sinh

ffiffiffig

p ~Hc

eth39THORN



frac14Ja

ks~Hcb


2ethY 2 ~HcTHORN

2 2b ~H

2

c

6

(

thorn 2X1nfrac141


sncos

npethY 2 ~HcTHORN

~Hc

esnX

)eth40THORN

in which


2

c

bthorn n 2p 2= ~H2

c


R frac14DT


TethL 0THORN2 T0


Ja

ks~Hcb


3b ~H

2

c

1






2ksrsaquoTs

rsaquozfrac14 2kf

rsaquoT f



T fethxTHORN frac14

Z wc=2

0


with

HFF151

16

um frac14

Z wc=2

0

v dz=wc=2


rsaquoT

rsaquoy

yfrac14H c

frac14rsaquoT

rsaquox

xfrac140

frac14rsaquoT

rsaquox

xfrac14L

frac14rsaquoT

rsaquoz

zfrac142ww

2

frac14rsaquoT f

rsaquoz

zfrac14wc

2

frac14 0 eth46THORN


ww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2

T thorn

rsaquoT

rsaquoz

0

2ww2

frac14 0 eth47THORN

in which

T frac14

Z 0

2ww2

T dz=ethww=2THORN


ksww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2



ksww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2




rsaquo

rsaquox

Z wc=2

0

uT f dz frac14kf

rfCp

rsaquoT f

rsaquoz

wc2

0

eth50THORN


umwc

2

rsaquo

rsaquoxT f thorn

1




DTceth52THORN

in which


17

DTc frac142hfa

2hfathorn 1


hH cq00 eth53THORN


A 2 rsaquo2

rsaquoX 2thorn

rsaquo2

rsaquoY 2

T 2 ethmH cTHORN


rsaquoT f


rsaquoT

rsaquoY

Yfrac140


rsaquoTs

rsaquoX

Xfrac141

frac14rsaquoTs

rsaquoY

Yfrac140

frac14rsaquoTs

rsaquoY

Yfrac141

frac14 0 eth57THORN



S frac14 hL

rfCpumwc

2







ifrac141

C0i

w0i


with

w01 frac14 2S

2thorn


2

2

thornmH c

A

2s

w02 frac14 2S

22


2

2

thornmH c

A

2s



HFF151

18



C03 frac14SA

mH c

2


R frac14DT


Teth1 0THORN2 T0

q00ethLW THORNDTc

frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


X2

ifrac141

C0i

w0i

ew0i thorn C03

1

ethLW THORN

eth61THORN


R frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


SA

mH c

2

1




2d

dxkplf thorn mf

d2

dy 2kulf 2

mf





U frac14 Dad2U

dY 22 P eth65THORN



U frac14kulf

um Da frac14

K

1H 2frac14

1

12a2s

Y frac14y

H P frac14

K

1mfum

dkplf

dx



19

U frac14 P cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 2 1

8gtltgt

9gt=gt eth67THORN




1rfCpkulf

rsaquokTlf


rsaquo2kTlf

rsaquoy 2eth69THORN



rsaquokTlrsaquoy

frac14rsaquokTlf





rsaquokTlf

rsaquoxfrac14

rsaquokTlrsaquox

frac14dTw


and


rsaquoxeth73THORN


d2u



dY 2eth75THORN

with


du

dYfrac14

duf


in which

HFF151

20

C frac141kf


haH 2


kTl2 Tw

q00Heth121THORNks

uf frac14kTlf 2 Tw

q00Heth121THORNks


uf frac14P

1 thorn C

242

1



Deth1 thorn CTHORN

CY

r 1A

2 C4 sinh


CY

r

thorn C5 cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt375

eth78THORN

u frac14 P

24Da cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt

21

2Y 2 thorn C1Y 2 Da

352 Cuf

eth79THORN

where



Da1 2 cosh

ffiffiffiffiffiffi1

Da

r ( )vuut

N 2 frac14C

Da


C

rsinh

ffiffiffiffiffiffi1

Da

r sinh


C

r

C1 frac14 1 2


pcosh

ffiffiffiffi1

Da

q 2 1

sinh

ffiffiffiffi1

Da

q


Deth1 thorn CTHORN

C3 frac14C



21


Deth1 thorn CTHORN


Ccosh


CY

q sinh

ffiffiffiffi1

Da

q D1

s

C5 frac14 Da 21

D1


R frac1412K

rfCpw3caW1 2um

2ufbH c



uf b frac14

Z 1

0

Uuf dY

13Z 1

0

U dy







HFF151

22

Tem

per

atu

reA

xia

lco

nd

uct

ion

Mod

elF

inF

luid

Fin

Flu

idG

over

nin

geq

uat

ion

sT

her

mal

resi

stan

ce(R

)

11D

1Dpound

poundN

otu

sed

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfQ

Cp

(88)

22D

1Dpound

pound

d2T

dy

22

hP

k sA

cethT

2T


0

_ mC

pdT

fethxTHORN

dx

frac14q

middotethw

cthorn

wwTHORN

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfC

pQ

(89)

32D

2Dpound

poundk srsaquo

2T

rsaquoy

2frac14

2h wwethT

2T

fethx

yTHORNTHORN

rfC

pu

mw

crsaquoT

f

rsaquox

frac142hethT

2T

fethx

yTHORNTHORN

Ja

k s~ H

cb

eth1thorn

L=aTHORNthorn

1 3b~ H

2 c

1

ethLW

THORN(9

0)

42D

2Dp

pound7

2TethxyzTHORNfrac14

0

7middotethk VT

fethx

yzTHORNTHORNfrac14

af7

2T

fethx

yzTHORN

2hfa

2hfa

thorn1

1

2nhH

ch

fLthorn

1

rfC

pQthorn

1

ethrfC

pQTHORN2

nk s

Hcw

w

L

(91)

51D

1Dpound

pound

2d dxkpl fthorn

mf

d2

dy

2kul f2

mf k1kul ffrac14

0

k sersaquo

2kT

lrsaquoy

2frac14

haethk

Tl2

kTl fTHORN

1r

fCpkul frsaquokT

l frsaquox

frac14haethk

Tl2

kTl fTHORNthorn

k fersaquo

2kT

l frsaquoy

2

12K

rfC

pw

3 ca

W1

2u

m2

uf

hH

c

eth12

1THORNk

sL

W(9

2)

Notespound

ndashn

otco

nsi

der

ed

andp

ndashco

nsi

der

ed


analytical models


23




f appRe frac1432

ethLthornTHORN057

2


1=2





Case




Case1 2 3

Nufd 597 581 606Nu 560 555 585


HFF151

24









rsaquoR


rsaquoR





1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072



25


References

















Further reading


HFF151

26

d2T

dy 2frac14

2h



2ksdT

dy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww

wwq00 eth20THORN

dT

dy

yfrac14H c

frac14 0 eth21THORN

It follows that


m

q00

ks

2hfa

2hfathorn 1

wc thorn ww

ww





_mCpdT fethxTHORN







m

q00

ks

2hfa

2hfathorn 1

wc thorn ww

ww






R frac14DT


TethL 0THORN2 T0

q00ethLW THORN

frac141

m

1

ks

2hfa

2hfathorn 1

wc thorn ww

ww

coshethmH cTHORN

sinhethmH cTHORN

1

ethLW THORNthorn

ethwc thorn wwTHORN

rfCpumH cwc

1

Weth26THORN


rsaquo2T

rsaquoy 2frac14

2h


with

HFF151

14

2ksrsaquoT

rsaquoy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww


rsaquoT

rsaquoy

yfrac14H c

frac14 0 eth29THORN


rfCpumwcrsaquoT f



1

2rfCpumwcDh

rsaquoT f




Z X

0





Z X

0


eth33THORN

in which

b frac14a2

l 2eth34THORN

l 2 frac14kswwDh

2kfNueth35THORN




bs

s thorn 1

eth36THORN


2ksrsaquof

rsaquoY

Yfrac140

frac14Ja

seth37THORN

2ksrsaquof

rsaquoY

Yfrac14 ~Hc

frac14 0 eth38THORN



15


kssffiffiffig

pcosh

ffiffiffig

pethY 2 ~HcTHORN

sinh

ffiffiffig

p ~Hc

eth39THORN



frac14Ja

ks~Hcb


2ethY 2 ~HcTHORN

2 2b ~H

2

c

6

(

thorn 2X1nfrac141


sncos

npethY 2 ~HcTHORN

~Hc

esnX

)eth40THORN

in which


2

c

bthorn n 2p 2= ~H2

c


R frac14DT


TethL 0THORN2 T0


Ja

ks~Hcb


3b ~H

2

c

1






2ksrsaquoTs

rsaquozfrac14 2kf

rsaquoT f



T fethxTHORN frac14

Z wc=2

0


with

HFF151

16

um frac14

Z wc=2

0

v dz=wc=2


rsaquoT

rsaquoy

yfrac14H c

frac14rsaquoT

rsaquox

xfrac140

frac14rsaquoT

rsaquox

xfrac14L

frac14rsaquoT

rsaquoz

zfrac142ww

2

frac14rsaquoT f

rsaquoz

zfrac14wc

2

frac14 0 eth46THORN


ww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2

T thorn

rsaquoT

rsaquoz

0

2ww2

frac14 0 eth47THORN

in which

T frac14

Z 0

2ww2

T dz=ethww=2THORN


ksww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2



ksww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2




rsaquo

rsaquox

Z wc=2

0

uT f dz frac14kf

rfCp

rsaquoT f

rsaquoz

wc2

0

eth50THORN


umwc

2

rsaquo

rsaquoxT f thorn

1




DTceth52THORN

in which


17

DTc frac142hfa

2hfathorn 1


hH cq00 eth53THORN


A 2 rsaquo2

rsaquoX 2thorn

rsaquo2

rsaquoY 2

T 2 ethmH cTHORN


rsaquoT f


rsaquoT

rsaquoY

Yfrac140


rsaquoTs

rsaquoX

Xfrac141

frac14rsaquoTs

rsaquoY

Yfrac140

frac14rsaquoTs

rsaquoY

Yfrac141

frac14 0 eth57THORN



S frac14 hL

rfCpumwc

2







ifrac141

C0i

w0i


with

w01 frac14 2S

2thorn


2

2

thornmH c

A

2s

w02 frac14 2S

22


2

2

thornmH c

A

2s



HFF151

18



C03 frac14SA

mH c

2


R frac14DT


Teth1 0THORN2 T0

q00ethLW THORNDTc

frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


X2

ifrac141

C0i

w0i

ew0i thorn C03

1

ethLW THORN

eth61THORN


R frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


SA

mH c

2

1




2d

dxkplf thorn mf

d2

dy 2kulf 2

mf





U frac14 Dad2U

dY 22 P eth65THORN



U frac14kulf

um Da frac14

K

1H 2frac14

1

12a2s

Y frac14y

H P frac14

K

1mfum

dkplf

dx



19

U frac14 P cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 2 1

8gtltgt

9gt=gt eth67THORN




1rfCpkulf

rsaquokTlf


rsaquo2kTlf

rsaquoy 2eth69THORN



rsaquokTlrsaquoy

frac14rsaquokTlf





rsaquokTlf

rsaquoxfrac14

rsaquokTlrsaquox

frac14dTw


and


rsaquoxeth73THORN


d2u



dY 2eth75THORN

with


du

dYfrac14

duf


in which

HFF151

20

C frac141kf


haH 2


kTl2 Tw

q00Heth121THORNks

uf frac14kTlf 2 Tw

q00Heth121THORNks


uf frac14P

1 thorn C

242

1



Deth1 thorn CTHORN

CY

r 1A

2 C4 sinh


CY

r

thorn C5 cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt375

eth78THORN

u frac14 P

24Da cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt

21

2Y 2 thorn C1Y 2 Da

352 Cuf

eth79THORN

where



Da1 2 cosh

ffiffiffiffiffiffi1

Da

r ( )vuut

N 2 frac14C

Da


C

rsinh

ffiffiffiffiffiffi1

Da

r sinh


C

r

C1 frac14 1 2


pcosh

ffiffiffiffi1

Da

q 2 1

sinh

ffiffiffiffi1

Da

q


Deth1 thorn CTHORN

C3 frac14C



21


Deth1 thorn CTHORN


Ccosh


CY

q sinh

ffiffiffiffi1

Da

q D1

s

C5 frac14 Da 21

D1


R frac1412K

rfCpw3caW1 2um

2ufbH c



uf b frac14

Z 1

0

Uuf dY

13Z 1

0

U dy







HFF151

22

Tem

per

atu

reA

xia

lco

nd

uct

ion

Mod

elF

inF

luid

Fin

Flu

idG

over

nin

geq

uat

ion

sT

her

mal

resi

stan

ce(R

)

11D

1Dpound

poundN

otu

sed

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfQ

Cp

(88)

22D

1Dpound

pound

d2T

dy

22

hP

k sA

cethT

2T


0

_ mC

pdT

fethxTHORN

dx

frac14q

middotethw

cthorn

wwTHORN

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfC

pQ

(89)

32D

2Dpound

poundk srsaquo

2T

rsaquoy

2frac14

2h wwethT

2T

fethx

yTHORNTHORN

rfC

pu

mw

crsaquoT

f

rsaquox

frac142hethT

2T

fethx

yTHORNTHORN

Ja

k s~ H

cb

eth1thorn

L=aTHORNthorn

1 3b~ H

2 c

1

ethLW

THORN(9

0)

42D

2Dp

pound7

2TethxyzTHORNfrac14

0

7middotethk VT

fethx

yzTHORNTHORNfrac14

af7

2T

fethx

yzTHORN

2hfa

2hfa

thorn1

1

2nhH

ch

fLthorn

1

rfC

pQthorn

1

ethrfC

pQTHORN2

nk s

Hcw

w

L

(91)

51D

1Dpound

pound

2d dxkpl fthorn

mf

d2

dy

2kul f2

mf k1kul ffrac14

0

k sersaquo

2kT

lrsaquoy

2frac14

haethk

Tl2

kTl fTHORN

1r

fCpkul frsaquokT

l frsaquox

frac14haethk

Tl2

kTl fTHORNthorn

k fersaquo

2kT

l frsaquoy

2

12K

rfC

pw

3 ca

W1

2u

m2

uf

hH

c

eth12

1THORNk

sL

W(9

2)

Notespound

ndashn

otco

nsi

der

ed

andp

ndashco

nsi

der

ed


analytical models


23




f appRe frac1432

ethLthornTHORN057

2


1=2





Case




Case1 2 3

Nufd 597 581 606Nu 560 555 585


HFF151

24









rsaquoR


rsaquoR





1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072



25


References

















Further reading


HFF151

26

2ksrsaquoT

rsaquoy

yfrac140

frac142hfa

2hfathorn 1

wc thorn ww


rsaquoT

rsaquoy

yfrac14H c

frac14 0 eth29THORN


rfCpumwcrsaquoT f



1

2rfCpumwcDh

rsaquoT f




Z X

0





Z X

0


eth33THORN

in which

b frac14a2

l 2eth34THORN

l 2 frac14kswwDh

2kfNueth35THORN




bs

s thorn 1

eth36THORN


2ksrsaquof

rsaquoY

Yfrac140

frac14Ja

seth37THORN

2ksrsaquof

rsaquoY

Yfrac14 ~Hc

frac14 0 eth38THORN



15


kssffiffiffig

pcosh

ffiffiffig

pethY 2 ~HcTHORN

sinh

ffiffiffig

p ~Hc

eth39THORN



frac14Ja

ks~Hcb


2ethY 2 ~HcTHORN

2 2b ~H

2

c

6

(

thorn 2X1nfrac141


sncos

npethY 2 ~HcTHORN

~Hc

esnX

)eth40THORN

in which


2

c

bthorn n 2p 2= ~H2

c


R frac14DT


TethL 0THORN2 T0


Ja

ks~Hcb


3b ~H

2

c

1






2ksrsaquoTs

rsaquozfrac14 2kf

rsaquoT f



T fethxTHORN frac14

Z wc=2

0


with

HFF151

16

um frac14

Z wc=2

0

v dz=wc=2


rsaquoT

rsaquoy

yfrac14H c

frac14rsaquoT

rsaquox

xfrac140

frac14rsaquoT

rsaquox

xfrac14L

frac14rsaquoT

rsaquoz

zfrac142ww

2

frac14rsaquoT f

rsaquoz

zfrac14wc

2

frac14 0 eth46THORN


ww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2

T thorn

rsaquoT

rsaquoz

0

2ww2

frac14 0 eth47THORN

in which

T frac14

Z 0

2ww2

T dz=ethww=2THORN


ksww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2



ksww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2




rsaquo

rsaquox

Z wc=2

0

uT f dz frac14kf

rfCp

rsaquoT f

rsaquoz

wc2

0

eth50THORN


umwc

2

rsaquo

rsaquoxT f thorn

1




DTceth52THORN

in which


17

DTc frac142hfa

2hfathorn 1


hH cq00 eth53THORN


A 2 rsaquo2

rsaquoX 2thorn

rsaquo2

rsaquoY 2

T 2 ethmH cTHORN


rsaquoT f


rsaquoT

rsaquoY

Yfrac140


rsaquoTs

rsaquoX

Xfrac141

frac14rsaquoTs

rsaquoY

Yfrac140

frac14rsaquoTs

rsaquoY

Yfrac141

frac14 0 eth57THORN



S frac14 hL

rfCpumwc

2







ifrac141

C0i

w0i


with

w01 frac14 2S

2thorn


2

2

thornmH c

A

2s

w02 frac14 2S

22


2

2

thornmH c

A

2s



HFF151

18



C03 frac14SA

mH c

2


R frac14DT


Teth1 0THORN2 T0

q00ethLW THORNDTc

frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


X2

ifrac141

C0i

w0i

ew0i thorn C03

1

ethLW THORN

eth61THORN


R frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


SA

mH c

2

1




2d

dxkplf thorn mf

d2

dy 2kulf 2

mf





U frac14 Dad2U

dY 22 P eth65THORN



U frac14kulf

um Da frac14

K

1H 2frac14

1

12a2s

Y frac14y

H P frac14

K

1mfum

dkplf

dx



19

U frac14 P cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 2 1

8gtltgt

9gt=gt eth67THORN




1rfCpkulf

rsaquokTlf


rsaquo2kTlf

rsaquoy 2eth69THORN



rsaquokTlrsaquoy

frac14rsaquokTlf





rsaquokTlf

rsaquoxfrac14

rsaquokTlrsaquox

frac14dTw


and


rsaquoxeth73THORN


d2u



dY 2eth75THORN

with


du

dYfrac14

duf


in which

HFF151

20

C frac141kf


haH 2


kTl2 Tw

q00Heth121THORNks

uf frac14kTlf 2 Tw

q00Heth121THORNks


uf frac14P

1 thorn C

242

1



Deth1 thorn CTHORN

CY

r 1A

2 C4 sinh


CY

r

thorn C5 cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt375

eth78THORN

u frac14 P

24Da cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt

21

2Y 2 thorn C1Y 2 Da

352 Cuf

eth79THORN

where



Da1 2 cosh

ffiffiffiffiffiffi1

Da

r ( )vuut

N 2 frac14C

Da


C

rsinh

ffiffiffiffiffiffi1

Da

r sinh


C

r

C1 frac14 1 2


pcosh

ffiffiffiffi1

Da

q 2 1

sinh

ffiffiffiffi1

Da

q


Deth1 thorn CTHORN

C3 frac14C



21


Deth1 thorn CTHORN


Ccosh


CY

q sinh

ffiffiffiffi1

Da

q D1

s

C5 frac14 Da 21

D1


R frac1412K

rfCpw3caW1 2um

2ufbH c



uf b frac14

Z 1

0

Uuf dY

13Z 1

0

U dy







HFF151

22

Tem

per

atu

reA

xia

lco

nd

uct

ion

Mod

elF

inF

luid

Fin

Flu

idG

over

nin

geq

uat

ion

sT

her

mal

resi

stan

ce(R

)

11D

1Dpound

poundN

otu

sed

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfQ

Cp

(88)

22D

1Dpound

pound

d2T

dy

22

hP

k sA

cethT

2T


0

_ mC

pdT

fethxTHORN

dx

frac14q

middotethw

cthorn

wwTHORN

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfC

pQ

(89)

32D

2Dpound

poundk srsaquo

2T

rsaquoy

2frac14

2h wwethT

2T

fethx

yTHORNTHORN

rfC

pu

mw

crsaquoT

f

rsaquox

frac142hethT

2T

fethx

yTHORNTHORN

Ja

k s~ H

cb

eth1thorn

L=aTHORNthorn

1 3b~ H

2 c

1

ethLW

THORN(9

0)

42D

2Dp

pound7

2TethxyzTHORNfrac14

0

7middotethk VT

fethx

yzTHORNTHORNfrac14

af7

2T

fethx

yzTHORN

2hfa

2hfa

thorn1

1

2nhH

ch

fLthorn

1

rfC

pQthorn

1

ethrfC

pQTHORN2

nk s

Hcw

w

L

(91)

51D

1Dpound

pound

2d dxkpl fthorn

mf

d2

dy

2kul f2

mf k1kul ffrac14

0

k sersaquo

2kT

lrsaquoy

2frac14

haethk

Tl2

kTl fTHORN

1r

fCpkul frsaquokT

l frsaquox

frac14haethk

Tl2

kTl fTHORNthorn

k fersaquo

2kT

l frsaquoy

2

12K

rfC

pw

3 ca

W1

2u

m2

uf

hH

c

eth12

1THORNk

sL

W(9

2)

Notespound

ndashn

otco

nsi

der

ed

andp

ndashco

nsi

der

ed


analytical models


23




f appRe frac1432

ethLthornTHORN057

2


1=2





Case




Case1 2 3

Nufd 597 581 606Nu 560 555 585


HFF151

24









rsaquoR


rsaquoR





1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072



25


References

















Further reading


HFF151

26


kssffiffiffig

pcosh

ffiffiffig

pethY 2 ~HcTHORN

sinh

ffiffiffig

p ~Hc

eth39THORN



frac14Ja

ks~Hcb


2ethY 2 ~HcTHORN

2 2b ~H

2

c

6

(

thorn 2X1nfrac141


sncos

npethY 2 ~HcTHORN

~Hc

esnX

)eth40THORN

in which


2

c

bthorn n 2p 2= ~H2

c


R frac14DT


TethL 0THORN2 T0


Ja

ks~Hcb


3b ~H

2

c

1






2ksrsaquoTs

rsaquozfrac14 2kf

rsaquoT f



T fethxTHORN frac14

Z wc=2

0


with

HFF151

16

um frac14

Z wc=2

0

v dz=wc=2


rsaquoT

rsaquoy

yfrac14H c

frac14rsaquoT

rsaquox

xfrac140

frac14rsaquoT

rsaquox

xfrac14L

frac14rsaquoT

rsaquoz

zfrac142ww

2

frac14rsaquoT f

rsaquoz

zfrac14wc

2

frac14 0 eth46THORN


ww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2

T thorn

rsaquoT

rsaquoz

0

2ww2

frac14 0 eth47THORN

in which

T frac14

Z 0

2ww2

T dz=ethww=2THORN


ksww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2



ksww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2




rsaquo

rsaquox

Z wc=2

0

uT f dz frac14kf

rfCp

rsaquoT f

rsaquoz

wc2

0

eth50THORN


umwc

2

rsaquo

rsaquoxT f thorn

1




DTceth52THORN

in which


17

DTc frac142hfa

2hfathorn 1


hH cq00 eth53THORN


A 2 rsaquo2

rsaquoX 2thorn

rsaquo2

rsaquoY 2

T 2 ethmH cTHORN


rsaquoT f


rsaquoT

rsaquoY

Yfrac140


rsaquoTs

rsaquoX

Xfrac141

frac14rsaquoTs

rsaquoY

Yfrac140

frac14rsaquoTs

rsaquoY

Yfrac141

frac14 0 eth57THORN



S frac14 hL

rfCpumwc

2







ifrac141

C0i

w0i


with

w01 frac14 2S

2thorn


2

2

thornmH c

A

2s

w02 frac14 2S

22


2

2

thornmH c

A

2s



HFF151

18



C03 frac14SA

mH c

2


R frac14DT


Teth1 0THORN2 T0

q00ethLW THORNDTc

frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


X2

ifrac141

C0i

w0i

ew0i thorn C03

1

ethLW THORN

eth61THORN


R frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


SA

mH c

2

1




2d

dxkplf thorn mf

d2

dy 2kulf 2

mf





U frac14 Dad2U

dY 22 P eth65THORN



U frac14kulf

um Da frac14

K

1H 2frac14

1

12a2s

Y frac14y

H P frac14

K

1mfum

dkplf

dx



19

U frac14 P cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 2 1

8gtltgt

9gt=gt eth67THORN




1rfCpkulf

rsaquokTlf


rsaquo2kTlf

rsaquoy 2eth69THORN



rsaquokTlrsaquoy

frac14rsaquokTlf





rsaquokTlf

rsaquoxfrac14

rsaquokTlrsaquox

frac14dTw


and


rsaquoxeth73THORN


d2u



dY 2eth75THORN

with


du

dYfrac14

duf


in which

HFF151

20

C frac141kf


haH 2


kTl2 Tw

q00Heth121THORNks

uf frac14kTlf 2 Tw

q00Heth121THORNks


uf frac14P

1 thorn C

242

1



Deth1 thorn CTHORN

CY

r 1A

2 C4 sinh


CY

r

thorn C5 cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt375

eth78THORN

u frac14 P

24Da cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt

21

2Y 2 thorn C1Y 2 Da

352 Cuf

eth79THORN

where



Da1 2 cosh

ffiffiffiffiffiffi1

Da

r ( )vuut

N 2 frac14C

Da


C

rsinh

ffiffiffiffiffiffi1

Da

r sinh


C

r

C1 frac14 1 2


pcosh

ffiffiffiffi1

Da

q 2 1

sinh

ffiffiffiffi1

Da

q


Deth1 thorn CTHORN

C3 frac14C



21


Deth1 thorn CTHORN


Ccosh


CY

q sinh

ffiffiffiffi1

Da

q D1

s

C5 frac14 Da 21

D1


R frac1412K

rfCpw3caW1 2um

2ufbH c



uf b frac14

Z 1

0

Uuf dY

13Z 1

0

U dy







HFF151

22

Tem

per

atu

reA

xia

lco

nd

uct

ion

Mod

elF

inF

luid

Fin

Flu

idG

over

nin

geq

uat

ion

sT

her

mal

resi

stan

ce(R

)

11D

1Dpound

poundN

otu

sed

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfQ

Cp

(88)

22D

1Dpound

pound

d2T

dy

22

hP

k sA

cethT

2T


0

_ mC

pdT

fethxTHORN

dx

frac14q

middotethw

cthorn

wwTHORN

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfC

pQ

(89)

32D

2Dpound

poundk srsaquo

2T

rsaquoy

2frac14

2h wwethT

2T

fethx

yTHORNTHORN

rfC

pu

mw

crsaquoT

f

rsaquox

frac142hethT

2T

fethx

yTHORNTHORN

Ja

k s~ H

cb

eth1thorn

L=aTHORNthorn

1 3b~ H

2 c

1

ethLW

THORN(9

0)

42D

2Dp

pound7

2TethxyzTHORNfrac14

0

7middotethk VT

fethx

yzTHORNTHORNfrac14

af7

2T

fethx

yzTHORN

2hfa

2hfa

thorn1

1

2nhH

ch

fLthorn

1

rfC

pQthorn

1

ethrfC

pQTHORN2

nk s

Hcw

w

L

(91)

51D

1Dpound

pound

2d dxkpl fthorn

mf

d2

dy

2kul f2

mf k1kul ffrac14

0

k sersaquo

2kT

lrsaquoy

2frac14

haethk

Tl2

kTl fTHORN

1r

fCpkul frsaquokT

l frsaquox

frac14haethk

Tl2

kTl fTHORNthorn

k fersaquo

2kT

l frsaquoy

2

12K

rfC

pw

3 ca

W1

2u

m2

uf

hH

c

eth12

1THORNk

sL

W(9

2)

Notespound

ndashn

otco

nsi

der

ed

andp

ndashco

nsi

der

ed


analytical models


23




f appRe frac1432

ethLthornTHORN057

2


1=2





Case




Case1 2 3

Nufd 597 581 606Nu 560 555 585


HFF151

24









rsaquoR


rsaquoR





1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072



25


References

















Further reading


HFF151

26

um frac14

Z wc=2

0

v dz=wc=2


rsaquoT

rsaquoy

yfrac14H c

frac14rsaquoT

rsaquox

xfrac140

frac14rsaquoT

rsaquox

xfrac14L

frac14rsaquoT

rsaquoz

zfrac142ww

2

frac14rsaquoT f

rsaquoz

zfrac14wc

2

frac14 0 eth46THORN


ww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2

T thorn

rsaquoT

rsaquoz

0

2ww2

frac14 0 eth47THORN

in which

T frac14

Z 0

2ww2

T dz=ethww=2THORN


ksww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2



ksww

2

rsaquo2

rsaquox 2thorn

rsaquo2

rsaquoy 2




rsaquo

rsaquox

Z wc=2

0

uT f dz frac14kf

rfCp

rsaquoT f

rsaquoz

wc2

0

eth50THORN


umwc

2

rsaquo

rsaquoxT f thorn

1




DTceth52THORN

in which


17

DTc frac142hfa

2hfathorn 1


hH cq00 eth53THORN


A 2 rsaquo2

rsaquoX 2thorn

rsaquo2

rsaquoY 2

T 2 ethmH cTHORN


rsaquoT f


rsaquoT

rsaquoY

Yfrac140


rsaquoTs

rsaquoX

Xfrac141

frac14rsaquoTs

rsaquoY

Yfrac140

frac14rsaquoTs

rsaquoY

Yfrac141

frac14 0 eth57THORN



S frac14 hL

rfCpumwc

2







ifrac141

C0i

w0i


with

w01 frac14 2S

2thorn


2

2

thornmH c

A

2s

w02 frac14 2S

22


2

2

thornmH c

A

2s



HFF151

18



C03 frac14SA

mH c

2


R frac14DT


Teth1 0THORN2 T0

q00ethLW THORNDTc

frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


X2

ifrac141

C0i

w0i

ew0i thorn C03

1

ethLW THORN

eth61THORN


R frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


SA

mH c

2

1




2d

dxkplf thorn mf

d2

dy 2kulf 2

mf





U frac14 Dad2U

dY 22 P eth65THORN



U frac14kulf

um Da frac14

K

1H 2frac14

1

12a2s

Y frac14y

H P frac14

K

1mfum

dkplf

dx



19

U frac14 P cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 2 1

8gtltgt

9gt=gt eth67THORN




1rfCpkulf

rsaquokTlf


rsaquo2kTlf

rsaquoy 2eth69THORN



rsaquokTlrsaquoy

frac14rsaquokTlf





rsaquokTlf

rsaquoxfrac14

rsaquokTlrsaquox

frac14dTw


and


rsaquoxeth73THORN


d2u



dY 2eth75THORN

with


du

dYfrac14

duf


in which

HFF151

20

C frac141kf


haH 2


kTl2 Tw

q00Heth121THORNks

uf frac14kTlf 2 Tw

q00Heth121THORNks


uf frac14P

1 thorn C

242

1



Deth1 thorn CTHORN

CY

r 1A

2 C4 sinh


CY

r

thorn C5 cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt375

eth78THORN

u frac14 P

24Da cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt

21

2Y 2 thorn C1Y 2 Da

352 Cuf

eth79THORN

where



Da1 2 cosh

ffiffiffiffiffiffi1

Da

r ( )vuut

N 2 frac14C

Da


C

rsinh

ffiffiffiffiffiffi1

Da

r sinh


C

r

C1 frac14 1 2


pcosh

ffiffiffiffi1

Da

q 2 1

sinh

ffiffiffiffi1

Da

q


Deth1 thorn CTHORN

C3 frac14C



21


Deth1 thorn CTHORN


Ccosh


CY

q sinh

ffiffiffiffi1

Da

q D1

s

C5 frac14 Da 21

D1


R frac1412K

rfCpw3caW1 2um

2ufbH c



uf b frac14

Z 1

0

Uuf dY

13Z 1

0

U dy







HFF151

22

Tem

per

atu

reA

xia

lco

nd

uct

ion

Mod

elF

inF

luid

Fin

Flu

idG

over

nin

geq

uat

ion

sT

her

mal

resi

stan

ce(R

)

11D

1Dpound

poundN

otu

sed

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfQ

Cp

(88)

22D

1Dpound

pound

d2T

dy

22

hP

k sA

cethT

2T


0

_ mC

pdT

fethxTHORN

dx

frac14q

middotethw

cthorn

wwTHORN

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfC

pQ

(89)

32D

2Dpound

poundk srsaquo

2T

rsaquoy

2frac14

2h wwethT

2T

fethx

yTHORNTHORN

rfC

pu

mw

crsaquoT

f

rsaquox

frac142hethT

2T

fethx

yTHORNTHORN

Ja

k s~ H

cb

eth1thorn

L=aTHORNthorn

1 3b~ H

2 c

1

ethLW

THORN(9

0)

42D

2Dp

pound7

2TethxyzTHORNfrac14

0

7middotethk VT

fethx

yzTHORNTHORNfrac14

af7

2T

fethx

yzTHORN

2hfa

2hfa

thorn1

1

2nhH

ch

fLthorn

1

rfC

pQthorn

1

ethrfC

pQTHORN2

nk s

Hcw

w

L

(91)

51D

1Dpound

pound

2d dxkpl fthorn

mf

d2

dy

2kul f2

mf k1kul ffrac14

0

k sersaquo

2kT

lrsaquoy

2frac14

haethk

Tl2

kTl fTHORN

1r

fCpkul frsaquokT

l frsaquox

frac14haethk

Tl2

kTl fTHORNthorn

k fersaquo

2kT

l frsaquoy

2

12K

rfC

pw

3 ca

W1

2u

m2

uf

hH

c

eth12

1THORNk

sL

W(9

2)

Notespound

ndashn

otco

nsi

der

ed

andp

ndashco

nsi

der

ed


analytical models


23




f appRe frac1432

ethLthornTHORN057

2


1=2





Case




Case1 2 3

Nufd 597 581 606Nu 560 555 585


HFF151

24









rsaquoR


rsaquoR





1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072



25


References

















Further reading


HFF151

26

DTc frac142hfa

2hfathorn 1


hH cq00 eth53THORN


A 2 rsaquo2

rsaquoX 2thorn

rsaquo2

rsaquoY 2

T 2 ethmH cTHORN


rsaquoT f


rsaquoT

rsaquoY

Yfrac140


rsaquoTs

rsaquoX

Xfrac141

frac14rsaquoTs

rsaquoY

Yfrac140

frac14rsaquoTs

rsaquoY

Yfrac141

frac14 0 eth57THORN



S frac14 hL

rfCpumwc

2







ifrac141

C0i

w0i


with

w01 frac14 2S

2thorn


2

2

thornmH c

A

2s

w02 frac14 2S

22


2

2

thornmH c

A

2s



HFF151

18



C03 frac14SA

mH c

2


R frac14DT


Teth1 0THORN2 T0

q00ethLW THORNDTc

frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


X2

ifrac141

C0i

w0i

ew0i thorn C03

1

ethLW THORN

eth61THORN


R frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


SA

mH c

2

1




2d

dxkplf thorn mf

d2

dy 2kulf 2

mf





U frac14 Dad2U

dY 22 P eth65THORN



U frac14kulf

um Da frac14

K

1H 2frac14

1

12a2s

Y frac14y

H P frac14

K

1mfum

dkplf

dx



19

U frac14 P cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 2 1

8gtltgt

9gt=gt eth67THORN




1rfCpkulf

rsaquokTlf


rsaquo2kTlf

rsaquoy 2eth69THORN



rsaquokTlrsaquoy

frac14rsaquokTlf





rsaquokTlf

rsaquoxfrac14

rsaquokTlrsaquox

frac14dTw


and


rsaquoxeth73THORN


d2u



dY 2eth75THORN

with


du

dYfrac14

duf


in which

HFF151

20

C frac141kf


haH 2


kTl2 Tw

q00Heth121THORNks

uf frac14kTlf 2 Tw

q00Heth121THORNks


uf frac14P

1 thorn C

242

1



Deth1 thorn CTHORN

CY

r 1A

2 C4 sinh


CY

r

thorn C5 cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt375

eth78THORN

u frac14 P

24Da cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt

21

2Y 2 thorn C1Y 2 Da

352 Cuf

eth79THORN

where



Da1 2 cosh

ffiffiffiffiffiffi1

Da

r ( )vuut

N 2 frac14C

Da


C

rsinh

ffiffiffiffiffiffi1

Da

r sinh


C

r

C1 frac14 1 2


pcosh

ffiffiffiffi1

Da

q 2 1

sinh

ffiffiffiffi1

Da

q


Deth1 thorn CTHORN

C3 frac14C



21


Deth1 thorn CTHORN


Ccosh


CY

q sinh

ffiffiffiffi1

Da

q D1

s

C5 frac14 Da 21

D1


R frac1412K

rfCpw3caW1 2um

2ufbH c



uf b frac14

Z 1

0

Uuf dY

13Z 1

0

U dy







HFF151

22

Tem

per

atu

reA

xia

lco

nd

uct

ion

Mod

elF

inF

luid

Fin

Flu

idG

over

nin

geq

uat

ion

sT

her

mal

resi

stan

ce(R

)

11D

1Dpound

poundN

otu

sed

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfQ

Cp

(88)

22D

1Dpound

pound

d2T

dy

22

hP

k sA

cethT

2T


0

_ mC

pdT

fethxTHORN

dx

frac14q

middotethw

cthorn

wwTHORN

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfC

pQ

(89)

32D

2Dpound

poundk srsaquo

2T

rsaquoy

2frac14

2h wwethT

2T

fethx

yTHORNTHORN

rfC

pu

mw

crsaquoT

f

rsaquox

frac142hethT

2T

fethx

yTHORNTHORN

Ja

k s~ H

cb

eth1thorn

L=aTHORNthorn

1 3b~ H

2 c

1

ethLW

THORN(9

0)

42D

2Dp

pound7

2TethxyzTHORNfrac14

0

7middotethk VT

fethx

yzTHORNTHORNfrac14

af7

2T

fethx

yzTHORN

2hfa

2hfa

thorn1

1

2nhH

ch

fLthorn

1

rfC

pQthorn

1

ethrfC

pQTHORN2

nk s

Hcw

w

L

(91)

51D

1Dpound

pound

2d dxkpl fthorn

mf

d2

dy

2kul f2

mf k1kul ffrac14

0

k sersaquo

2kT

lrsaquoy

2frac14

haethk

Tl2

kTl fTHORN

1r

fCpkul frsaquokT

l frsaquox

frac14haethk

Tl2

kTl fTHORNthorn

k fersaquo

2kT

l frsaquoy

2

12K

rfC

pw

3 ca

W1

2u

m2

uf

hH

c

eth12

1THORNk

sL

W(9

2)

Notespound

ndashn

otco

nsi

der

ed

andp

ndashco

nsi

der

ed


analytical models


23




f appRe frac1432

ethLthornTHORN057

2


1=2





Case




Case1 2 3

Nufd 597 581 606Nu 560 555 585


HFF151

24









rsaquoR


rsaquoR





1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072



25


References

















Further reading


HFF151

26



C03 frac14SA

mH c

2


R frac14DT


Teth1 0THORN2 T0

q00ethLW THORNDTc

frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


X2

ifrac141

C0i

w0i

ew0i thorn C03

1

ethLW THORN

eth61THORN


R frac142hfa

2hfathorn 1

wc thorn ww

2hH cmH c

coshethmH cTHORN


SA

mH c

2

1




2d

dxkplf thorn mf

d2

dy 2kulf 2

mf





U frac14 Dad2U

dY 22 P eth65THORN



U frac14kulf

um Da frac14

K

1H 2frac14

1

12a2s

Y frac14y

H P frac14

K

1mfum

dkplf

dx



19

U frac14 P cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 2 1

8gtltgt

9gt=gt eth67THORN




1rfCpkulf

rsaquokTlf


rsaquo2kTlf

rsaquoy 2eth69THORN



rsaquokTlrsaquoy

frac14rsaquokTlf





rsaquokTlf

rsaquoxfrac14

rsaquokTlrsaquox

frac14dTw


and


rsaquoxeth73THORN


d2u



dY 2eth75THORN

with


du

dYfrac14

duf


in which

HFF151

20

C frac141kf


haH 2


kTl2 Tw

q00Heth121THORNks

uf frac14kTlf 2 Tw

q00Heth121THORNks


uf frac14P

1 thorn C

242

1



Deth1 thorn CTHORN

CY

r 1A

2 C4 sinh


CY

r

thorn C5 cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt375

eth78THORN

u frac14 P

24Da cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt

21

2Y 2 thorn C1Y 2 Da

352 Cuf

eth79THORN

where



Da1 2 cosh

ffiffiffiffiffiffi1

Da

r ( )vuut

N 2 frac14C

Da


C

rsinh

ffiffiffiffiffiffi1

Da

r sinh


C

r

C1 frac14 1 2


pcosh

ffiffiffiffi1

Da

q 2 1

sinh

ffiffiffiffi1

Da

q


Deth1 thorn CTHORN

C3 frac14C



21


Deth1 thorn CTHORN


Ccosh


CY

q sinh

ffiffiffiffi1

Da

q D1

s

C5 frac14 Da 21

D1


R frac1412K

rfCpw3caW1 2um

2ufbH c



uf b frac14

Z 1

0

Uuf dY

13Z 1

0

U dy







HFF151

22

Tem

per

atu

reA

xia

lco

nd

uct

ion

Mod

elF

inF

luid

Fin

Flu

idG

over

nin

geq

uat

ion

sT

her

mal

resi

stan

ce(R

)

11D

1Dpound

poundN

otu

sed

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfQ

Cp

(88)

22D

1Dpound

pound

d2T

dy

22

hP

k sA

cethT

2T


0

_ mC

pdT

fethxTHORN

dx

frac14q

middotethw

cthorn

wwTHORN

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfC

pQ

(89)

32D

2Dpound

poundk srsaquo

2T

rsaquoy

2frac14

2h wwethT

2T

fethx

yTHORNTHORN

rfC

pu

mw

crsaquoT

f

rsaquox

frac142hethT

2T

fethx

yTHORNTHORN

Ja

k s~ H

cb

eth1thorn

L=aTHORNthorn

1 3b~ H

2 c

1

ethLW

THORN(9

0)

42D

2Dp

pound7

2TethxyzTHORNfrac14

0

7middotethk VT

fethx

yzTHORNTHORNfrac14

af7

2T

fethx

yzTHORN

2hfa

2hfa

thorn1

1

2nhH

ch

fLthorn

1

rfC

pQthorn

1

ethrfC

pQTHORN2

nk s

Hcw

w

L

(91)

51D

1Dpound

pound

2d dxkpl fthorn

mf

d2

dy

2kul f2

mf k1kul ffrac14

0

k sersaquo

2kT

lrsaquoy

2frac14

haethk

Tl2

kTl fTHORN

1r

fCpkul frsaquokT

l frsaquox

frac14haethk

Tl2

kTl fTHORNthorn

k fersaquo

2kT

l frsaquoy

2

12K

rfC

pw

3 ca

W1

2u

m2

uf

hH

c

eth12

1THORNk

sL

W(9

2)

Notespound

ndashn

otco

nsi

der

ed

andp

ndashco

nsi

der

ed


analytical models


23




f appRe frac1432

ethLthornTHORN057

2


1=2





Case




Case1 2 3

Nufd 597 581 606Nu 560 555 585


HFF151

24









rsaquoR


rsaquoR





1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072



25


References

















Further reading


HFF151

26

U frac14 P cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 2 1

8gtltgt

9gt=gt eth67THORN




1rfCpkulf

rsaquokTlf


rsaquo2kTlf

rsaquoy 2eth69THORN



rsaquokTlrsaquoy

frac14rsaquokTlf





rsaquokTlf

rsaquoxfrac14

rsaquokTlrsaquox

frac14dTw


and


rsaquoxeth73THORN


d2u



dY 2eth75THORN

with


du

dYfrac14

duf


in which

HFF151

20

C frac141kf


haH 2


kTl2 Tw

q00Heth121THORNks

uf frac14kTlf 2 Tw

q00Heth121THORNks


uf frac14P

1 thorn C

242

1



Deth1 thorn CTHORN

CY

r 1A

2 C4 sinh


CY

r

thorn C5 cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt375

eth78THORN

u frac14 P

24Da cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt

21

2Y 2 thorn C1Y 2 Da

352 Cuf

eth79THORN

where



Da1 2 cosh

ffiffiffiffiffiffi1

Da

r ( )vuut

N 2 frac14C

Da


C

rsinh

ffiffiffiffiffiffi1

Da

r sinh


C

r

C1 frac14 1 2


pcosh

ffiffiffiffi1

Da

q 2 1

sinh

ffiffiffiffi1

Da

q


Deth1 thorn CTHORN

C3 frac14C



21


Deth1 thorn CTHORN


Ccosh


CY

q sinh

ffiffiffiffi1

Da

q D1

s

C5 frac14 Da 21

D1


R frac1412K

rfCpw3caW1 2um

2ufbH c



uf b frac14

Z 1

0

Uuf dY

13Z 1

0

U dy







HFF151

22

Tem

per

atu

reA

xia

lco

nd

uct

ion

Mod

elF

inF

luid

Fin

Flu

idG

over

nin

geq

uat

ion

sT

her

mal

resi

stan

ce(R

)

11D

1Dpound

poundN

otu

sed

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfQ

Cp

(88)

22D

1Dpound

pound

d2T

dy

22

hP

k sA

cethT

2T


0

_ mC

pdT

fethxTHORN

dx

frac14q

middotethw

cthorn

wwTHORN

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfC

pQ

(89)

32D

2Dpound

poundk srsaquo

2T

rsaquoy

2frac14

2h wwethT

2T

fethx

yTHORNTHORN

rfC

pu

mw

crsaquoT

f

rsaquox

frac142hethT

2T

fethx

yTHORNTHORN

Ja

k s~ H

cb

eth1thorn

L=aTHORNthorn

1 3b~ H

2 c

1

ethLW

THORN(9

0)

42D

2Dp

pound7

2TethxyzTHORNfrac14

0

7middotethk VT

fethx

yzTHORNTHORNfrac14

af7

2T

fethx

yzTHORN

2hfa

2hfa

thorn1

1

2nhH

ch

fLthorn

1

rfC

pQthorn

1

ethrfC

pQTHORN2

nk s

Hcw

w

L

(91)

51D

1Dpound

pound

2d dxkpl fthorn

mf

d2

dy

2kul f2

mf k1kul ffrac14

0

k sersaquo

2kT

lrsaquoy

2frac14

haethk

Tl2

kTl fTHORN

1r

fCpkul frsaquokT

l frsaquox

frac14haethk

Tl2

kTl fTHORNthorn

k fersaquo

2kT

l frsaquoy

2

12K

rfC

pw

3 ca

W1

2u

m2

uf

hH

c

eth12

1THORNk

sL

W(9

2)

Notespound

ndashn

otco

nsi

der

ed

andp

ndashco

nsi

der

ed


analytical models


23




f appRe frac1432

ethLthornTHORN057

2


1=2





Case




Case1 2 3

Nufd 597 581 606Nu 560 555 585


HFF151

24









rsaquoR


rsaquoR





1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072



25


References

















Further reading


HFF151

26

C frac141kf


haH 2


kTl2 Tw

q00Heth121THORNks

uf frac14kTlf 2 Tw

q00Heth121THORNks


uf frac14P

1 thorn C

242

1



Deth1 thorn CTHORN

CY

r 1A

2 C4 sinh


CY

r

thorn C5 cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt375

eth78THORN

u frac14 P

24Da cosh


DaY

r thorn


Da

q sinh

ffiffiffiffi1

Da

q sinh


DaY

r 8gtltgt

9gt=gt

21

2Y 2 thorn C1Y 2 Da

352 Cuf

eth79THORN

where



Da1 2 cosh

ffiffiffiffiffiffi1

Da

r ( )vuut

N 2 frac14C

Da


C

rsinh

ffiffiffiffiffiffi1

Da

r sinh


C

r

C1 frac14 1 2


pcosh

ffiffiffiffi1

Da

q 2 1

sinh

ffiffiffiffi1

Da

q


Deth1 thorn CTHORN

C3 frac14C



21


Deth1 thorn CTHORN


Ccosh


CY

q sinh

ffiffiffiffi1

Da

q D1

s

C5 frac14 Da 21

D1


R frac1412K

rfCpw3caW1 2um

2ufbH c



uf b frac14

Z 1

0

Uuf dY

13Z 1

0

U dy







HFF151

22

Tem

per

atu

reA

xia

lco

nd

uct

ion

Mod

elF

inF

luid

Fin

Flu

idG

over

nin

geq

uat

ion

sT

her

mal

resi

stan

ce(R

)

11D

1Dpound

poundN

otu

sed

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfQ

Cp

(88)

22D

1Dpound

pound

d2T

dy

22

hP

k sA

cethT

2T


0

_ mC

pdT

fethxTHORN

dx

frac14q

middotethw

cthorn

wwTHORN

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfC

pQ

(89)

32D

2Dpound

poundk srsaquo

2T

rsaquoy

2frac14

2h wwethT

2T

fethx

yTHORNTHORN

rfC

pu

mw

crsaquoT

f

rsaquox

frac142hethT

2T

fethx

yTHORNTHORN

Ja

k s~ H

cb

eth1thorn

L=aTHORNthorn

1 3b~ H

2 c

1

ethLW

THORN(9

0)

42D

2Dp

pound7

2TethxyzTHORNfrac14

0

7middotethk VT

fethx

yzTHORNTHORNfrac14

af7

2T

fethx

yzTHORN

2hfa

2hfa

thorn1

1

2nhH

ch

fLthorn

1

rfC

pQthorn

1

ethrfC

pQTHORN2

nk s

Hcw

w

L

(91)

51D

1Dpound

pound

2d dxkpl fthorn

mf

d2

dy

2kul f2

mf k1kul ffrac14

0

k sersaquo

2kT

lrsaquoy

2frac14

haethk

Tl2

kTl fTHORN

1r

fCpkul frsaquokT

l frsaquox

frac14haethk

Tl2

kTl fTHORNthorn

k fersaquo

2kT

l frsaquoy

2

12K

rfC

pw

3 ca

W1

2u

m2

uf

hH

c

eth12

1THORNk

sL

W(9

2)

Notespound

ndashn

otco

nsi

der

ed

andp

ndashco

nsi

der

ed


analytical models


23




f appRe frac1432

ethLthornTHORN057

2


1=2





Case




Case1 2 3

Nufd 597 581 606Nu 560 555 585


HFF151

24









rsaquoR


rsaquoR





1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072



25


References

















Further reading


HFF151

26


Deth1 thorn CTHORN


Ccosh


CY

q sinh

ffiffiffiffi1

Da

q D1

s

C5 frac14 Da 21

D1


R frac1412K

rfCpw3caW1 2um

2ufbH c



uf b frac14

Z 1

0

Uuf dY

13Z 1

0

U dy







HFF151

22

Tem

per

atu

reA

xia

lco

nd

uct

ion

Mod

elF

inF

luid

Fin

Flu

idG

over

nin

geq

uat

ion

sT

her

mal

resi

stan

ce(R

)

11D

1Dpound

poundN

otu

sed

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfQ

Cp

(88)

22D

1Dpound

pound

d2T

dy

22

hP

k sA

cethT

2T


0

_ mC

pdT

fethxTHORN

dx

frac14q

middotethw

cthorn

wwTHORN

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfC

pQ

(89)

32D

2Dpound

poundk srsaquo

2T

rsaquoy

2frac14

2h wwethT

2T

fethx

yTHORNTHORN

rfC

pu

mw

crsaquoT

f

rsaquox

frac142hethT

2T

fethx

yTHORNTHORN

Ja

k s~ H

cb

eth1thorn

L=aTHORNthorn

1 3b~ H

2 c

1

ethLW

THORN(9

0)

42D

2Dp

pound7

2TethxyzTHORNfrac14

0

7middotethk VT

fethx

yzTHORNTHORNfrac14

af7

2T

fethx

yzTHORN

2hfa

2hfa

thorn1

1

2nhH

ch

fLthorn

1

rfC

pQthorn

1

ethrfC

pQTHORN2

nk s

Hcw

w

L

(91)

51D

1Dpound

pound

2d dxkpl fthorn

mf

d2

dy

2kul f2

mf k1kul ffrac14

0

k sersaquo

2kT

lrsaquoy

2frac14

haethk

Tl2

kTl fTHORN

1r

fCpkul frsaquokT

l frsaquox

frac14haethk

Tl2

kTl fTHORNthorn

k fersaquo

2kT

l frsaquoy

2

12K

rfC

pw

3 ca

W1

2u

m2

uf

hH

c

eth12

1THORNk

sL

W(9

2)

Notespound

ndashn

otco

nsi

der

ed

andp

ndashco

nsi

der

ed


analytical models


23




f appRe frac1432

ethLthornTHORN057

2


1=2





Case




Case1 2 3

Nufd 597 581 606Nu 560 555 585


HFF151

24









rsaquoR


rsaquoR





1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072



25


References

















Further reading


HFF151

26

Tem

per

atu

reA

xia

lco

nd

uct

ion

Mod

elF

inF

luid

Fin

Flu

idG

over

nin

geq

uat

ion

sT

her

mal

resi

stan

ce(R

)

11D

1Dpound

poundN

otu

sed

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfQ

Cp

(88)

22D

1Dpound

pound

d2T

dy

22

hP

k sA

cethT

2T


0

_ mC

pdT

fethxTHORN

dx

frac14q

middotethw

cthorn

wwTHORN

1

nhLeth2h

fHcthorn

wcTHORNthorn

1

rfC

pQ

(89)

32D

2Dpound

poundk srsaquo

2T

rsaquoy

2frac14

2h wwethT

2T

fethx

yTHORNTHORN

rfC

pu

mw

crsaquoT

f

rsaquox

frac142hethT

2T

fethx

yTHORNTHORN

Ja

k s~ H

cb

eth1thorn

L=aTHORNthorn

1 3b~ H

2 c

1

ethLW

THORN(9

0)

42D

2Dp

pound7

2TethxyzTHORNfrac14

0

7middotethk VT

fethx

yzTHORNTHORNfrac14

af7

2T

fethx

yzTHORN

2hfa

2hfa

thorn1

1

2nhH

ch

fLthorn

1

rfC

pQthorn

1

ethrfC

pQTHORN2

nk s

Hcw

w

L

(91)

51D

1Dpound

pound

2d dxkpl fthorn

mf

d2

dy

2kul f2

mf k1kul ffrac14

0

k sersaquo

2kT

lrsaquoy

2frac14

haethk

Tl2

kTl fTHORN

1r

fCpkul frsaquokT

l frsaquox

frac14haethk

Tl2

kTl fTHORNthorn

k fersaquo

2kT

l frsaquoy

2

12K

rfC

pw

3 ca

W1

2u

m2

uf

hH

c

eth12

1THORNk

sL

W(9

2)

Notespound

ndashn

otco

nsi

der

ed

andp

ndashco

nsi

der

ed


analytical models


23




f appRe frac1432

ethLthornTHORN057

2


1=2





Case




Case1 2 3

Nufd 597 581 606Nu 560 555 585


HFF151

24









rsaquoR


rsaquoR





1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072



25


References

















Further reading


HFF151

26




f appRe frac1432

ethLthornTHORN057

2


1=2





Case




Case1 2 3

Nufd 597 581 606Nu 560 555 585


HFF151

24









rsaquoR


rsaquoR





1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072



25


References

















Further reading


HFF151

26









rsaquoR


rsaquoR





1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072



25


References

















Further reading


HFF151

26


References

















Further reading


HFF151

26

Date post:	26-Jun-2020
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