HEAT TRANSFER HANDBOOK doc

KRAUS University of Akron Akron, Ohio 1.1 Heat transfer fundamentals1.1.1 Introduction1.1.2 Conduction heat transferOne-dimensional conductionOne-dimensional conduction with internal hea

Adrian Bejan

J A Jones Professor of Mechanical EngineeringDepartment of Mechanical Engineering

Duke UniversityDurham, North CarolinaAllan D KrausDepartment of Mechanical EngineeringUniversity of Akron

Akron, Ohio

JOHN WILEY & SONS, INC.

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Library of Congress Cataloging-in-Publication Data:

Bejan, Adrian, 1948–

Heat transfer handbook / Adrian Bejan, Allan D Kraus.

p cm.

ISBN 0-471-39015-1 (cloth : alk paper)

1 Heat—Transmission—Handbooks, manuals, etc I Kraus, Allan D II Title.

TJ250 B35 2003

Printed in the United States of America

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PREFACE

Heat transfer has emerged as a central discipline in contemporary engineering ence The research activity of a few decades ago—the material reviewed in the firsthandbooks—has distilled itself into textbook concepts and results Heat transfer hasbecome not only a self-standing discipline in the current literature and engineeringcurricula, but also an indispensable discipline at the interface with other pivotal andolder disciplines For example, fluid mechanics today is capable of describing thetransport of heat and other contaminants because of the great progress made in mod-ern convective heat transfer Thermodynamics today is able to teach modeling, sim-ulation, and optimization of “realistic” energy systems because of the great progressmade in heat transfer Ducts, extended surfaces, heat exchangers, and other featuresthat may be contemplated by the practitioner are now documented in the heat transferliterature

sci-To bring this body of results to the fingertips of the reader is one of the objectives

of this new handbook The more important objective, however, is to inform the reader

on what has been happening in the field more recently In brief, heat transfer marchesforward through new ideas, applications, and emerging technologies The vigor ofheat transfer has always come from its usefulness For example, the challenges ofenergy self-sufficiency and aerospace travel, which moved the field in the 1970s,are still with us; in fact, they are making a strong comeback Another example isthe miniaturization revolution, which continues unabated The small-scale channels

of the 1980s do not look so small anymore Even before “small scale” became thefashion, we in heat transfer had “compact” heat exchangers The direction for thefuture is clear

The importance of optimizing the architecture of a flow system to make it fit into

a finite volume with purpose has always been recognized in heat transfer It has beenand continues to be the driving force Space comes at a premium Better and bettershapes of extended surfaces are evolving into networks, bushes, and trees of fins Themany surfaces designed for heat transfer augmentation are accomplishing the same

thing: They are increasing the heat transfer rate density, the size of the heat transfer

enterprise that is packed into a given volume

The smallest features are becoming smaller, but this is only half of the story Theother is the march toward greater complexity More and more small-scale featuresmust be connected and assembled into a device whose specified size is always macro-scopic Small-scale technologies demand the optimization of increasingly complexheat-flow architectures

A highly distinguished group of colleagues who are world authorities on thefrontiers of heat transfer today have contributed to this new handbook Their chaptersprovide a bird’s-eye view of the state of the field, highlighting both the foundations

One feature of the handbook is that the main results and correlations are rized at the ends of chapters This feature was chosen to provide quick access and

summa-to help the flow of heat transfer knowledge from research summa-to computer-aided design

It is our hope that researchers and practitioners of heat transfer will find this newhandbook inspiring and useful

Adrian Bejan acknowledges with gratitude the support received from ProfessorKristina Johnson, Dean of the Pratt School of Engineering, and Professor KennethHall, Chairman of the Department of Mechanical Engineering and Materials Science,Duke University Allan Kraus acknowledges the assistance of his wife, who hashelped in the proofreading stage of production

Both authors acknowledge the assistance of our editor at John Wiley, Bob tieri, our production editor, Milagros Torres, and our fantastic copy editor, knownonly to us as Barbara from Pennsylvania

Argen-Adrian BejanAllan D Kraus

Carnegie Mellon UniversityPittsburgh, PA 15213-3980Benjamin T F Chung

F Theodore Harrington EmeritusProfessor

Department of MechanicalEngineering

302 East Buchtel MallUniversity of AkronAkron, OH 44325-3903Avram Bar-CohenProfessor and ChairDepartment of MechanicalEngineering

2181B Martin HallUniversity of MarylandCollege Park, MD 20742-3035

Sadik KakacDepartment of MechanicalEngineering

University of MiamiCoral Gables, FL 33124-0624

G P PetersonProvostRensselaer Polytechnic Institute

110 Eighth StreetTroy, NY 12180-3590James Welty

Department of MechanicalEngineering

Rogers HallOregon State UniversityCorvallis, OR 97330Michael M YovanovichDepartment of MechanicalEngineering

University of WaterlooWaterloo, Ontario N2L 3G1Canada

Current address: Glenn L Martin Institute of Technology, A James Clark School

of Engineering, Department of Mechanical Engineering, 2181 Glenn L MartinHall, College Park, MD 20742-3035

Adrian Bejan, J A Jones Professor of Mechanical Engineering, Department of

Me-chanical Engineering and Materials Science, Duke University, Durham, NC 0300

27708-Robert F Boehm, University of Nevada–Las Vegas, Las Vegas, NV 89154-4027

J C Chato, Department of Mechanical and Industrial Engineering, University of

Yogesh Jaluria, Mechanical and Aerospace Engineering Department, Rutgers

Uni-versity, New Brunswick, NJ 08901-1281

Yogendra Joshi, George W Woodruff School of Mechanical Engineering, Georgia

Institute of Technology, Atlanta, GA 30332-0405

M A Kedzierski, Building and Fire Research Laboratory, National Institute of

Standards and Technology, Gaithersburg, MD 20899

Allan D Kraus, University of Akron, Akron, OH 44325-3901 José L Lage, Laboratory of Porous Materials Applications, Mechanical Engineer-

ing Department, Southern Methodist University, Dallas, TX 75275-0337

E W Lemmon, Physical and Chemical Properties Division, National Institute of

Standards and Technology, Boulder, CO 80395-3328

R M Manglik, Thermal-Fluids and Thermal Processing Laboratory, Department

of Mechanical, Industrial and Nuclear Engineering, University of Cincinnati, 598Rhodes Hall, P.O Box 210072, Cincinnati, OH 45221-0072

xiii

xiv CONTRIBUTORS

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E E Marotta, Senior Engineer/Scientist, Thermal Technologies Group, IBM

Cor-poration, Poughkeepsie, NY 12801

Michael F Modest, Professor of Mechanical and Nuclear Engineering, College of

Engineering, Pennsylvania State University, University Park, PA 16802-1412

Wataru Nakayama, Therm Tech International, Kanagawa, Japan 255-0004 Pamela M Norris, Associate Professor, Department of Mechanical and Aerospace

Engineering, University of Virginia, Charlottesville, VA 22903

Jay M Ochterbeck, College of Engineering and Science, Department of

Mechani-cal Engineering, Clemson University, Clemson, SC 29634-0921

S G Penoncello, Center for Applied Thermodynamic Studies, College of

Engineer-ing, University of Idaho, Moscow, ID 83844-1011

Ranga Pitchumani, Department of Mechanical Engineering, University of

Con-necticut, Storrs, CT 06269-3139

Ravi S Prasher, Intel Corporation, Chandler, AZ 85225

T J Rabas, Consultant, Downers Grove, IL 60516

Z Shan, Center for Applied Thermodynamic Studies, College of Engineering,

Uni-versity of Idaho, Moscow, ID 83844-1011

Andrew N Smith, Department of Mechanical Engineering, United States Naval

Academy, Annapolis, MD 21402-5000

Richard N Smith, Department of Mechanical Engineering, Aeronautical

Engineer-ing and Mechanics, Rensselaer Polytechnic Institute, Troy, NY 12180-3590

John R Thome, Laboratory of Heat and Mass Transfer, Faculty of Engineering

Sci-ence, Swiss Federal Institute of Technology Lausanne, CH-1015 Lausanne, land

Switzer-Abhay A Watwe, Intel Corporation, Chandler, AZ 85225

N T Wright, Department of Mechanical Engineering, University of Maryland,

Baltimore, MD 21250

M M Yovanovich, Distinguished Professor Emeritus, Department of Mechanical

Engineering, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

Preface ix

Allan D Kraus

R T Jacobsen, E W Lemmon, S G Penoncello, Z Shan, and N T Wright

Yogendra Joshi and Wataru Nakayama

Avram Bar-Cohen, Abhay A Watwe, and Ravi S Prasher

R M Manglik

vii

15 Porous Media 1131

Adrian Bejan

Jay M Ochterbeck

17 Heat Transfer in Manufacturing and Materials Processing 1231

Richard N Smith, C Haris Doumanidis, and Ranga Pitchumani

Andrew N Smithand Pamela M Norris

Robert F Boehm

ALLAN D KRAUS

University of Akron Akron, Ohio

1.1 Heat transfer fundamentals1.1.1 Introduction1.1.2 Conduction heat transferOne-dimensional conductionOne-dimensional conduction with internal heat generation1.1.3 Spreading resistance

1.1.4 Interface–contact resistance1.1.5 Lumped-capacity heating and cooling1.1.6 Convective heat transfer

Heat transfer coefficientDimensionless parametersNatural convectionForced convection1.1.7 Phase-change heat transfer1.1.8 Finned surfaces

1.1.9 Flow resistance1.1.10 Radiative heat transfer1.2 Coordinate systems

1.2.1 Rectangular (Cartesian) coordinate system1.2.2 Cylindrical coordinate system

1.2.3 Spherical coordinate system1.2.4 General curvilinear coordinates1.3 Continuity equation

1.4 Momentum and the momentum theorem1.5 Conservation of energy

1.6 Dimensional analysis1.6.1 Friction loss in pipe flow1.6.2 Summary of dimensionless groups1.7 Units

1.7.1 SI system (Syst`eme International d’Unit´es)1.7.2 English engineering system (U.S customary system)1.7.3 Conversion factors

NomenclatureReferences

1

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1.1 HEAT TRANSFER FUNDAMENTALS 1.1.1 Introduction

Practitioners of the thermal arts and sciences generally deal with four basic thermaltransport modes: conduction, convection, phase change, and radiation The process

by which heat diffuses through a solid or a stationary fluid is termed heat conduction.

Situations in which heat transfer from a wetted surface is assisted by the motion of

the fluid give rise to heat convection, and when the fluid undergoes a liquid–solid

or liquid–vapor state transformation at or very near the wetted surface, attention is

focused on this phase-change heat transfer The exchange of heat between surfaces,

or between a surface and a surrounding fluid, by long-wavelength electromagnetic

radiation is termed thermal heat radiation.

It is our intent in this section to describe briefly these modes of heat transfer, with

emphasis on an important parameter known as the thermal resistance to heat transfer.

Simple examples are given for illustration; detailed descriptions of the same topicsare presented in specialized chapters

1.1.2 Conduction Heat Transfer

by Fourier’s law, which in one-dimensional form is expressible as

whereq is the heat current, k the thermal conductivity of the medium, A the

cross-sectional area for heat flow, anddT /dx the temperature gradient, which, because it

is negative, requires insertion of the minus sign in eq (1.1) to assure a positive heatflowq The temperature difference resulting from the steady-state diffusion of heat

is thus related to the thermal conductivity of the material, the cross-sectional areaA,

and the path lengthL (Fig 1.1), according to

The form of eq (1.2), wherek and A are presumed constant, suggests that in a way

that is analogous to Ohm’s law governing electrical current flow through a resistance,

it is possible to define a conduction thermal resistance as

R cd≡ T1 − T2

L

in which a solid experiences internal heat generation, such as that produced by theflow of an electric current, give rise to more complex governing equations and require

T

x

Figure 1.1 Heat transfer by conduction through a slab

greater care in obtaining the appropriate temperature differences The axial ature variation in the slim, internally heated conductor shown in Fig 1.2 is found toequal

2

whereT o is the edge temperature When the two ends are cooled to an identicaltemperature, and when the volumetric heat generation rateq g (W/m3) is uniformthroughout, the peak temperature is developed at the center of the solid and is givenby

Tmax = T o + q g L2

Alternatively, becauseq g is the volumetric heat generation q g = q/LWδ, the

center–edge temperature difference can be expressed as

Tmax − T o = q L2

8kLWδ = q

L

where the cross-sectional areaA is the product of the width W and the thickness δ.

An examination of eq (1.5) reveals that the thermal resistance of a conductor with adistributed heat input is only one-fourth that of a structure in which all of the heat isgenerated at the center

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in successive “layers” in the conducting medium below the source The additional

resistance associated with this lateral flow of heat is called the spreading resistance.

According to Yovanovich and Antonetti (1988), the spreading resistance for a smallheat source on a thick conductor or heat spreader (required to be three to five timesthicker than the square root of the heat source area) can be expressed as

Rsp= 1− 1.410 + 0.3443+ 0.0435+ 0.0347

where  is the ratio of the heat source area to the substrate area, k the thermal

conductivity of the conductor, anda the square root of the area of the heat source.

For relatively thin conducting layers on thicker substrates, such as encountered

in the cooling of microcircuits, eq (1.6) cannot provide an acceptable prediction of

Rsp Instead, use can be made of the numerical results plotted in Fig 1.3 to obtain therequisite value of the spreading resistance

1.1.4 Interface–Contact Resistance

Heat transfer across the interface between two solids is generally accompanied by

a measurable temperature difference, which can be ascribed to a contact or face thermal resistance For perfectly adhering solids, geometrical differences in thecrystal structure (lattice mismatch) can impede the flow of phonons and electrons

across the interface, but this resistance is generally negligible in engineering design

However, when dealing with real interfaces, the asperities present on each of the faces (Fig 1.4) limit actual contact between the two solids to a very small fraction

sur-of the apparent interface area The flow sur-of heat across the gap between two solids innominal contact is by solid conduction in areas of actual contact and fluid conductionacross the “open” spaces Radiation across the gap can be important in a vacuumenvironment or when surface temperatures are high The heat transferred across aninterface can be found by adding the effects of solid-to-solid conduction and conduc-tion through the fluid and recognizing that solid-to-solid conduction in the contactzones involves heat flowing sequentially through the two solids With the total con-tact conductanceh co, taken as the sum of solid-to-solid conductanceh cand the gapconductanceh g,

0.95

(1.8a)

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Figure 1.4 Physical contact between two nonideal surfaces

wherek sis the harmonic mean thermal conductivity for solid 1 and solid 2,

P is the contact pressure, and H is the microhardness of the softer material, both in

N/m2 In the absence of detailed information, theσ/m ratio can be taken as 5 to 9 µm

for relatively smooth surfaces

In eq (1.7a),h gis given by

wherek g is the thermal conductivity of the gap fluid,Y is the distance between the

mean planes (Fig 1.4), given by

0.547

andM is a gas parameter used to account for rarified gas effects,

M = αβΛ

whereα is an accommodation parameter (approximately equal to 1.7 for air and cleanmetals),Λ is the mean free path of the molecules (equal to approximately 0.06 µmfor air at atmospheric pressure and 15°C), andβ is a fluid property parameter (equal

to approximately 1.7 for air and other diatomic gases) Equations (1.8a) and (1.8b)can be added and, in accordance with eq (1.7a), the contact resistance becomes

R co= 1.25k s m

σ



P H

0.95+ k g

1.1.5 Lumped-Capacity Heating and Cooling

An internally heated solid of relatively high thermal conductivity that is experiencing

no external cooling will undergo a constant rise in temperature according to

dT

q

whereq is the rate of internal heat generation, m the mass of the solid, and c the

specific heat of the solid Equation (1.10) assumes that all the mass can be represented

by a single temperature This approach is commonly called the lumped-capacity model for transient heating.

Expanding on the analogy between thermal and electrical resistances suggestedpreviously, the product of mass and specific heat can be viewed as analogous to

electrical capacitance and thus to constitute the thermal capacitance.

When this same solid is externally cooled, the temperature rises asymptoticallytoward the steady-state temperature, which is itself determined by the external resis-tance to heat flow,R Consequently, the time variation of the temperature of the solid

is expressible as

where the product of the external resistanceR and the thermal capacitance mc is seen

to constitute the thermal time constant of the system.

1.1.6 Convective Heat Transfer

fluid in motion can be related to the heat tranfser coefficienth, the surface-to-fluid

temperature difference, and the “wetted” surface areaS in the form

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flow regime,h may be found from available empirical correlations and/or

theoret-ical relations Use of eq (1.12) makes it possible to define the convective thermalresistance as

R cv≡ 1

the correlation of heat transfer data are the Nusselt number Nu, which relates the

convective heat transfer coefficient to the conduction in the fluid:

hL k The Prandtl number Pr, which is a fluid property parameter:

Natural Convection In natural convection, fluid motion is induced by densitydifferences resulting from temperature gradients in the fluid The heat transfer coef-ficient for this regime can be related to the buoyancy and the thermal properties of

the fluid through the Rayleigh number Ra, which is the product of the Grashof and

Empirical correlations for the heat transfer coefficient in natural convection boundarylayer flow have taken the form

of the correlating coefficientC depends on fluid, the geometry of the surface, and

the Rayleigh number range Nevertheless, for common plate, cylinder, and sphereconfigurations, it is found to vary in the relatively narrow range of 0.45 to 0.65 forlaminar flow and 0.11 to 0.15 for turbulent flow past the heated surface

Natural convection in vertical channels such as those formed by arrays of gitudinal fins is of major significance Elenbaas (1942) was the first to document adetailed study of this configuration, and his experimental results for isothermal plateswere later confirmed numerically by Bodoia and Osterle (1964) A uniform picture

lon-of the thermal transport in such a vertical channel has emerged from these and plementary studies

com-It has been shown that the value of the Nusselt number lies between two extremesthat are based on the size of the space between the plates or width of channel Forwide spacing, the plates appear to have little influence on one another, and the Nusselt

number in this case achieves its isolated plate limit On the other hand, for closely spaced plates or for relatively long channels, the fluid attains its fully developed value and the Nusselt number reaches its fully developed limit Intermediate values of the

Nusselt number can be obtained from a correlating method suggested by Churchilland Usagi (1972) for smoothly varying processes, and these values have been verified

by a plethora of detailed experimental and numerical studies

Thus, the correlation for the average value ofh along isothermal vertical channels

spacedz units apart is

h = k z

576

Forced Convection For forced flow in long or very narrow parallel-plate nels, the heat transfer coefficient attains an asymptotic value (the fully developedlimit), which for symmetrically heated channel surfaces is equal approximately to

chan-h = 4k

whered e is the hydraulic diameter defined in terms of the flow area A and the wetted

perimeter of the surfacesp:

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In higher-velocity turbulent flow along plates, the dependence of the convectiveheat transfer coefficient on the Reynolds number increases, and in the range Re ≥

Re≤ 10,000 usually referred to as the transition region For the transition region,

Hausen (1943) has provided the correlating equation

1.1.7 Phase-Change Heat Transfer

Boiling heat transfer displays a complex dependence on the temperature differencebetween the heated surface and the saturation temperature (boiling point) of theliquid Following Rohsenow (1952), the heat transfer rate in nucleate boiling, theprimary region of interest, can be approximated by a relation of the form

whereC sf is a function of the surface–fluid combination For comparison purposes,

it is possible to define a boiling heat transfer coefficienthφ:

whereS f is the surface area of the fin,T b the temperature at the base of the fin,T s

the surrounding temperature, andq b the heat entering the base of the fin, which inthe steady state is equal to the heat dissipated by the fin The thermal resistance of afinned surface is given by

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Based on this relation, it is possible to define an effective flow resistance,R f l, as

R f l≡ 1

1.1.10 Radiative Heat Transfer

Unlike conduction and convection, radiative heat transfer between two surfaces orbetween a surface and its surroundings is not linearly dependent on the temperaturedifference and is expressed instead as

q = σA F

T4

1 − T4 2



whereF includes the effects of surface properties and geometry andσ is the Stefan–

Boltzmann constant,σ = 5.669 × 10−8W/m2· K4 For modest temperature ences, this equation can be linearized to the form

whereh ris the effective “radiation” heat transfer coefficient,

h r = σFT2

1 + T2 2

to the convective resistance, is seen to equal

x z

Origin

Figure 1.5 Rectangular (Cartesian), cylindrical, and spherical coordinate systems

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1.2.1 Rectangular (Cartesian) Coordinate System

For a rectangular coordinate system with coordinatesx,y, and z and unit vectors e x,

ey , and ez, the gradient of the scalarT is

1.2.2 Cylindrical Coordinate System

For a cylindrical coordinate system with coordinatesr, θ, and z and unit vectors e r

eθ, and ez, the gradient of the scalarT is

r2

∂2T

∂θ2 +∂2T

1.2.3 Spherical Coordinate System

For a spherical coordinate system with coordinatesr, θ, and φ and unit vectors e r, eθ,

and eφ, the gradient of the scalarT is

∂ dφ

sinφ∂T ∂φ

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1.2.4 General Curvilinear Coordinates

In general, a curvilinear coordinate system can be proposed where a vector V has

componentsV1, V2, andV3in thex1, x2, andx3coordinate directions The unit vectors

are e1, e2, and e3in the coordinate directionsx1, x2, andx3and there are scale factors,

s1, s2, ands3that relate the general curvilinear coordinate system to the rectangular,cylindrical, and spherical coordinate systems

In the general curvilinear coordinate system, the gradient of a scalarT is

The curl of the vector V is

The Laplacian of the scalarT is

∇2T = 1s1s2s3

can be applied to the control volume by noting that the net rate of mass flux out ofthe control volume plus the rate of accumulation of mass within the control volumemust equal zero:

and that the mass flux at each of the faces of Fig 1.6 will beρ( ˆV · n), where n is the

normal to the areadA.

Noting that the density can vary from point to point and with time,ρ = f (x,y,z, t), the net mass flux out of the control volumes in each of the coordinate directions

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where∇ · ρ ˆV = div ρ ˆV is the divergence of the vector ˆV This equation is general: It

applies to unsteady three-dimensional flow with variableρ

Equation (1.51) is a vector equation that represents the equation of continuity in

rectangular, cylindrical, and spherical coordinates If the flow is incompressible, sothatρ is independent of time, eq (1.51) reduces to

The momentum theorem of fluid mechanics provides a relation between a group offield points It is especially useful when the details of the flow field are more thanmoderately complicated and it is based on Newton’s law, which can be written as

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proper-At timet2, these particles will have moved to a region bounded by the control surface

S2, which is shown as a dashed curve to distinguish it fromS1.The control surfacesS1andS2enclose three separate and distinct regions, desig-nated bya,b, and c Let the momentum in the three regions be P a , P b, and Pc, re-spectively At timet1the particles within surfaceS1will possess momentum Pa+Pb1

At timet2these particles will have momentum, Pb2+ Pcbecause they have movedinto the region enclosed by surfaceS2 Hence the momentum change during the timeintervalt2 − t1may be described by

The second term in eq (1.65) is the momentum efflux through the control surface

S1 If the flux in the outward direction is taken as positive, this efflux can be expressed

c b a

Figure 1.7 Regions bounded by control surfaces used for the development of the momentumtheorem

The conservation of momentum principle then becomes

in the three rectangular coordinate directions

The foregoing development leads to the statement of the momentum theorem: The

time rate of increase of momentum of a fluid within a fixed control volumeR will

be equal to the rate at which momentum flows intoR through its confining surface

S, plus the net force acting on the fluid within R When the flow is incompressible, the viscosity is constant, and the flow is laminar, the Navier–Stokes equations result.

In Cartesian coordinates, withF x,F y, andF ztaken as the components of the bodyforce per unit volume, the Navier–Stokes equations are

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In cylindrical coordinates withF r Fθ, andF ztaken as the components of the body

force per unit volume, the Navier–Stokes equations are

r



= −∂P ∂r+ µ

In Fig 1.8, an imaginary two-dimensional control volume of finite size∆x ∆y with

flow velocity ˆV = ex ˆV x+ ey ˆV y, heat flux q= ex q

x+ ey q

y, specific internal energy

u, and rate of internal heat generation q, the first law of thermodynamics requiresthat

 rate of energyaccumulation withinthe control volume



=

 net transfer

of energy byfluid flow

+

 net heattransfer byconduction



×

 rate ofinternal heatgeneration

• The rate of energy accumulated in the control volume is

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• The rate of internal heat generation is

The origin of the term involving the net work transferred from the control volume

to the environment is shown in Fig 1.9, where the normal and tangential stresses aresketched For example, the work done per unit time by the normal stressσxon the left

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side of the∆x ∆y element is negative and equal to the force acting on the boundary

σy, multiplied by the boundary displacement per unit time ˆV x This yields− ˆV xσx∆y.

Similarly, the work transfer associated with normal stresses acting on the right side

of the element is positive and equal to

so that the work transfer becomes

whereµ is the dynamic viscosity of the fluid and Φ is the viscous dissipation function,

which is detailed subsequently in rectangular, cylindrical, and spherical coordinates

However, eq (1.53) shows that the term in parentheses on the left-hand side of eq

(1.74) is equal to zero, so that eq (1.74) reduces to

y can be expressed in terms of local temperature

gradients through use of Fourier’s law:

Here, too, eq (1.53) points out that the terms in parentheses in eq (1.79) are equal tozero, so that eq (1.75) reduces to

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P dT +



ds dP

generation (q = 0), and a negligible compressibility effect, βT (DP /Dt) ≈ 0 The

energy equation for this model is simply

in the cylindrical coordinate system,

∂θ2 +∂ ∂z2T2

(1.92)

and in the spherical coordinate system,

∂

∂φ

sinφ∂T ∂φ

If the fluid can be modeled as incompressible then, as in eq (1.89b), the specificheat at constant pressure c p is replaced by c And when dealing with extremely

viscous flows, the model is improved by taking into account the internal heating due



∂ ˆV y

∂y

2+