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Chapter 2

Thermodynamics, Fluid Dynamics,and Heat Transfer

2.1 Introduction

In this chapter we will review fundamental concepts from Thermodynamics, FluidDynamics, and Heat Transfer. Each section first begins with a review of the funda-mentals. Subsequently, a review of important equations and solutions to fundamentalproblems from each of the three fields. This chapter is only intended to provide thenecessary reference material for the course. It is not intended as a substitute for thebasic texts used in the thermo-fluids courses. During this course extensive referencewill be made to the basic texts dealing with thermo-fluid fundamentals.

Where possible, the use of robust design models or correlations which span a widerange of flow conditions will be encouraged. These comprehensive models allow forgreater flexibility in the design and optimization of thermal systems. Whereas piece-wise models, i.e. those which consider each different flow region separately, tend todetract from the integrated design approach developed in the notes.

2.2 Thermodynamics

2.2.1 First Law of Thermodynamics

The First Law of Thermodynamics, better known as the conservation of energy willbe utilized for both open and closed systems throughout this course. We shall beginby examining the different ways of stating the First Law for both open and closedsystems.

The First Law of Thermodynamics for a closed system states

E2 − E1 = Q1−2 + W1−2 (2.1)

This is understood to imply that the change in energy of a closed system is related

15

16 Mechanical Equipment and Systems

to the net heat input and the net work done on the system. In terms of instantaneoustransfer rates, the First Law may be written on a per unit time basis

dE

dt= Q + W (2.2)

The First Law of Thermodynamics may also be written for an open system con-taining a number of inlets and outlets

dEcv

dt= mi

(ui + pivi +

V 2i

2+ gzi

)− me

(ue + peve +

V 2e

2+ gze

)+ Qcv + Wcv (2.3)

This equation states that the accumulation of energy within the control volumemust equal the net inflow of energy into the control volume minus the net outflowof energy from the control volume plus the increase in energy due to work and heattransfers. The sign convention adopted in these notes is that any work done ona system is considered positive, while any work done by the system is considerednegative. This convention reflects the notion that work done on a system increasesthe energy of the system, i.e. the use of a pump or compressor. Contemporary textsin Thermodynamics have preferred the use of the “heat engine” convention whichreflects that useful work done by the system is considered positive. Either conventionmay be applied so long as consistency is applied throughout the analysis of a problem.

2.2.2 Second Law of Thermodynamics

The Second Law of Thermodynamics deals with the irreversibility of thermodynamicprocesses. A reversible process is one in which there is no production of entropy. Fora closed system the second law of thermodynamics states that∫ 2

1

δQ

T≤ S2 − S1 (2.4)

We define the entropy production Sgen as the difference between the entropy changeand the entropy transfer such that

Sgen = (S2 − S1)−∫ 2

1

δQ

T≥ 0 (2.5)

The Second Law of Thermodynamics for an open system containing a number ofinlets and outlets becomes

dScv

dt=∑ Qj

Tj

+∑

misi −∑

mese + Sgen (2.6)

This equation states that the rate of entropy accumulation within the control vol-ume is balanced by the net transfer of entropy through heat exchanges with thesurroundings plus the net flow of entropy into the control volume and the rate of

Fundamentals 17

entropy production within the control volume. It should be noted that for a steadystate analysis the entropy production rate is not zero (except for reversible processes).Whereas the rate of accumulation of entropy within the control volume is zero for asteady state process.

2.2.3 Exergy

The first and second laws of thermodynamics may be combined to develop a newrelation which governs a new quantity Exergy. Exergy is a measure of the potentialof a thermodynamic system to do work. Unlike energy, exergy can be destroyed.Exergy analysis, sometimes called availability analysis, is used quite frequently in thedesign and analysis of thermal systems. Exergy is defined as

E = (E − Uo) + po(V − Vo)− To(S − So) (2.7)

Here E = (U + P.E. + K.E.), the energy of the system, U is internal energy, Sis entropy, and V is the volume of the system. The reference or dead state as it isreferred is denoted by the subscript (·)o. We may also define exergy as an intensiveproperty, that is on a per unit mass basis, such that

e = [(u + V 2/2 + gz)− uo] + po(v − vo)− To(s− so) (2.8)

The change in exergy between any two states is merely

E2 − E1 = (E2 − E1) + po(V2 − V1)− To(S2 − S1) (2.9)

For a closed system the exergy balance yields

E2 − E1 =

∫ 2

1

(1− To

Tb

)δQ− [W − po(V2 − V1)]− Ed (2.10)

The term Ed = ToSgen, is the exergy which is destroyed due to irreversibilities in thesystem. For an open system with a number of inlets and outlets the exergy balanceyields:

dEcv

dt=∑(

1− To

Tj

)Qj −

(Wcv − po

dVcv

dt

)+∑

miei −∑

meee − Ed (2.11)

where

e = (h− ho)− To(s− so) + V 2/2 + gz (2.12)

is the flow exergy.


2.3 Dimensionless Groups

Before proceeding to the review of fluid dynamics and heat transfer models, a briefdiscussion on the use of dimensionless quantities is required. A number of importantdimensionless quantities appear throughout the text. The student should familiarizehimself or herself with these parameters and their use. Table 1 summarizes the mostimportant groups that will be encountered during this course.

Table 1Frequently Encountered Dimensionless Groups

Group Definition

Biot Number Bi =hLks

Reynolds Number Re =ρV L

µ

Prandtl Number Pr =ν

α

Peclet Number Pe =V Lα

= RePr

Grashof Number Gr =gβ∆TL3

ν2

Rayleigh Number Ra =gβ∆TL3

αν= GrPr

Nusselt Number Nu =(q/A)Lkf∆T

=hLkf

Stanton Number St =Nu

RePr

Colburn Factor j =Nu

RePr1/3

Friction Coefficient Cf =τ

12ρV 2

Fanning Friction Factor f =(∆p/L)(A/P )

12ρV 2

Fundamentals 19

2.4 Fluid Dynamics

2.4.1 Conservation Equations

Conservation of mass and momentum for a control volume will be applied throughoutthe course. Here we will merely state the general form as previously discussed in fluidmechanics courses.

Conservation of Mass

dmCV

dt=∑inlets

mi −∑exits

me (2.13)

Conservation of Momentum∑~Fext =

∑exits

~Ve(ρ~VeAe)−∑inlets

~Vi(ρ~ViAi) (2.14)

In addition, we will also apply Bernoulli’s equation for a number of incompressibleflows.

Bernoulli’s Equation

P1

γ+

V 21

2g+ z1 =

P2

γ+

V 22

2g+ z2 + hL (2.15)

2.4.2 Internal Flows

When analyzing flow in ducting or piping systems as well as flow through mechanicalequipment, a number of design models and correlations are required for relating themass flow rate to the pressure drop of the working fluid. The most common methodis through the definition of the friction factor. The Fanning friction factor will beadopted for this course. It is defined as follows:

f =τ

12ρu2

=

A

P

∆p

L12ρu2

=

Dh

4

∆p

L12ρu2

(2.16)

where

Dh =4A

P(2.17)

where A is the cross-sectional area and P is the perimeter of the duct. In fullydeveloped laminar flows the friction factor takes the following form:

f =C

ReDh

(2.18)


where C is a constant which is a function of the shape and aspect ratio of the duct.Table 2 summarizes a number of values for common duct shapes.

Apparent friction factors for developing flows may be computed from the followingformula

fappReDh=

[(3.44√

L∗

)2

+ (fReDh)2

]1/2

(2.19)

where

L∗ =L

DhReDh

(2.20)

In circular tubes the flow is developing in a region where L∗ < 0.058. The entrancelength for laminar flow development in a tube is:

Le = 0.058DReD (2.21)

Table 2Typical values of fReDh

= C forNon-Circular Ducts

Shape fReDh= C

Equilateral Triangle 13.33Square 14.23Pentagon 14.74Hexagon 15.05Octagon 15.41Circle 16Elliptic 2:1 16.82Elliptic 4:1 18.24Elliptic 8:1 19.15Rectangular 2:1 15.55Rectangular 4:1 18.23Rectangular 8:1 20.58Parallel Plates 24

For turbulent flows the friction factor is predicted using the Colebrook relation.This correlation is the basis for the Moody diagram. However, it is not a practicalequation since it defines the friction factor implicitly. An alternate form proposed bySwamee and Jain (1976) which provides accuracy within ±1.5% is given by

f =1

16

[log

(k/Dh

3.7+

5.74

Re9/10Dh

)]2 (2.22)

Fundamentals 21

The entrance length for turbulent flow in a tube is:

Le = 4.4D(ReD)1/6 (2.23)

Finally, in teh case of non-circular ducts we use the concept of the hydraulic diam-eter D = Dh = 4A/P defined earlier, to compute an equivalent duct diameter.

2.4.3 External Flows

A number of important design equations for external fluid flows are required to relatethe free stream velocity to the overall drag force. The three most common geometriesare the flat plate, the cylinder, and the sphere.

Flat PlateFor laminar boundary layer flows, 1000 < ReL < 500, 000, the important parame-

ters are the boundary layer thickness and the friction coefficient:

δ(x) =5x

Re1/2x

(2.24)

Cf,x =0.664

Re1/2x

(2.25)

Cf =1.328

Re1/2L

(2.26)

For turbulent boundary layer flows, 500, 000 < ReL < 107, the boundary layerthickness and friction coefficient are:

δ(x) =0.38x

Re1/5x

(2.27)

Cf,x =0.059

Re1/5x

(2.28)

Cf =0.074

Re1/5L

(2.29)

If the boundary layer is composed of a combined laminar-turbulent flow, ReL >500, 000, the friction coefficient is computed from the integrated value:

Cf =0.074

Re1/5L

− 1742

ReL

(2.30)


Finally, a number of useful models for predicting drag on flat plates, cylinders, andspheres in low Reynolds number flows are also provided. These models will providethe building blocks for analyzing a fluid component or system.

Flat Plate 0.01 < ReL < 500, 000

Cf =2.66

Re7/8L

+1.328

Re1/2L

(2.31)

Cylinder 0.1 < ReD < 250, 000

CD =10

Re2/3D

+ 1.0 (2.32)

Sphere 0.01 < ReD < 250, 000

CD =24

ReD

+6

1 + Re1/2D

+ 0.4 (2.33)

where

CD, Cf =F/A12ρu2

(2.34)

Note care must be taken to ensure the correct characteristic area A is chosen basedupon the geometry.

2.5 Heat Transfer

2.5.1 Conduction

1-Dimensional Steady Conduction

Steady one-dimensional conduction in plane walls, cylinders, and spheres is easilyanalyzed using the resistance concept. The thermal resistance is defined such that

∆T = QRt (2.35)

For a multi-component system containing j layers, the following thermal resistanceresults are useful.

Plane Wall

Rt =1

hiA+∑ tj

kjA+

1

hoA(2.36)

Fundamentals 23

Cylinder

Rt =1

(2πriL)hi

+∑ ln(roj/rij)

(2πkjL)+

1

(2πroL)ho

(2.37)

Sphere

Rt =1

(4πr2i )hi

+∑ 1

4πkj

(1

rij

− 1

roj

)+

1

(4πr2o)ho

(2.38)

Multi-Dimensional Steady Conduction

In two or three dimensions, heat transfer by means of conduction is best analyzedusing shape factors. Many multi-dimensional solutions of practical interest have beenobtained and are outlined below. The conduction shape factor S, is defined such that:

R =1

Sk(2.39)

where R is the thermal resistance and k is the thermal conductivity of the medium.Shape factors for several simple systems are given in Fig. 1. The shape factor S, isonly a function of the geometry of the system. The overall heat transfer rate is thenrelated to an appropriate temperature difference:

Q = Sk∆T (2.40)

where ∆T is the temperature difference between two isothermal surfaces or betweena surface and a remote sink.

Transient Conduction

Transient conduction in finite and semi-infinite regions are also of interest. Thefollowing solutions are useful for modelling a number of thermal systems.

Semi-Infinite Regions

Isothermal WallT (x, t)− Ts

Ti − Ts

= erf

(x

2√

αt

)(2.41)

qs(t) =k(Ts − Ti)√

παt(2.42)


Fig. 2.1a - Some Useful Shape Factors, From Handbook of Heat Transfer,Rohsenow et al., McGraw-Hill, 1985.

Fundamentals 25

Fig. 2.1b - Some Useful Shape Factors, From Handbook of Heat Transfer,Rohsenow et al., McGraw-Hill, 1985.


Fig. 2.1c - Some Useful Shape Factors, From Handbook of Heat Transfer,Rohsenow et al., McGraw-Hill, 1985.

Isoflux Wall

T (x, t)− Ti =2qs

√αt/π

kexp

(−x2

4αt

)− qsx

kerfc

(x

2√

αt

)(2.43)

Ts(t)− Ti =2qs

k

(αt

π

)1/2

(2.44)

Surface Convection

T (x, t)− Ti

T∞ − Ti

= erfc

(x

2√

αt

)−[exp

(hx

k+

h2αt

k2

)][erfc

(x

2√

αt+

h√

αt

k

)](2.45)

Ts(t)− Ti

T∞ − Ti

= 1− exp

(h2αt

k2

)erfc

(h√

αt

k

)(2.46)

qs(t)√

αt

k(T∞ − Ti)= exp

(h2αt

k2

)erfc

(h√

αt

k

)(2.47)

Fundamentals 27

Finite Regions

Transient conduction from finite one dimensional and multi-dimensional regionsmay be analyzed using the following solutions. In the solutions below θ = T − Tf ,θi = Ti − Tf , and Qi = ρcpV (Ti − Tf ). The notation adopted in this section followsthat of Yovanovich (1999).

Plane Wall

θ

θi

=∞∑

n=1

An exp(−δ2nFo) cos(δnX) (2.48)

where

An =4 sin(δn)

2δn + sin(2δn)(2.49)

The eigenvalues δn are determined from

δn sin(δn) = Bi cos(δn) (2.50)

In the expressions above, Fo = αt/L2, X = x/L, and Bi = hL/k. The heat flowat the surface of the wall is determined from

Q

Qi

= 1−∞∑

n=1

(2Bi2

δ2n(Bi2 + Bi + δ2

n)

)exp(−δ2

nFo) (2.51)

Next if Fo > 0.24, the series solutions for temperature and heat flow reduce tosingle term approximations

θ

θi

= A1 exp(−δ21Fo) cos(δ1X) (2.52)

Q

Qi

= 1−(

2Bi2

δ21(Bi2 + Bi + δ2

1)

)exp(−δ2

1Fo) (2.53)

where

δ1 =1.5708

[1 + (1.5708/√

Bi)2.139]0.4675(2.54)

Finally, if the Biot number is small (Bi < 0.2), spatial effects are no longer sig-nificant and the lumped capacitance model applies. For a plane wall this resultsin

θ

θi

= exp(−BiFo) (2.55)

Q

Qi

= 1− exp(−BiFo) (2.56)


Infinite Cylinder

θ

θi

=∞∑

n=1

An exp(−δ2nFo)J0(δnR) (2.57)

where

An =2J1(δn)

δn(J20 (δn) + J2

1 (δn))(2.58)


δnJ1(δn) = J0(δn)Bi (2.59)

In the expressions above, Fo = αt/a2, R = r/a, and Bi = ha/k. The heat flow atthe surface of the cylinder is determined from

Q

Qi

= 1−∞∑

n=1

(4Bi2

δ2n(Bi2 + δ2

n)

)exp(−δ2

nFo) (2.60)


θ

θi

= A1 exp(−δ21Fo)J0(δ1R) (2.61)

Q

Qi

= 1−(

4Bi2

δ21(Bi2 + δ2

1)

)exp(−δ2

1Fo) (2.62)

where

δ1 =2.4048

[1 + (2.4048/√

2Bi)2.238]0.4468(2.63)

Finally, if the Biot number is small (Bi < 0.2), spatial effects are no longer signifi-cant and the lumped capacitance model applies. For an infinite cylinder this resultsin

θ

θi

= exp(−2BiFo) (2.64)

Q

Qi

= 1− exp(−2BiFo) (2.65)

Sphere

θ

θi

=∞∑

n=1

An exp(−δ2nFo)

sin(δnR)

δnR(2.66)

Fundamentals 29

where

An =4[sin(δn)− δn cos(δn)]

2δn − sin(2δn)(2.67)


δn cos(δn) = (1−Bi) sin(δn) (2.68)

In the expressions above, Fo = αt/a2, R = a/L, and Bi = ha/k. The heat flow atthe surface of the sphere is determined from

Q

Qi

= 1−∞∑

n=1

(6Bi2

δ2n(Bi2 −Bi + δ2

n)

)exp(−δ2

nFo) (2.69)


θ

θi

= A1 exp(−δ21Fo)

sin(δ1R)

δ1R(2.70)

Q

Qi

= 1−(

6Bi2

δ21(Bi2 −Bi + δ2

1)

)exp(−δ2

1Fo) (2.71)

where

δ1 =3.14159

[1 + (3.14159/√

3Bi)2.314]0.4322(2.72)

Finally, if the Biot number is small (Bi < 0.2), spatial effects are no longer signifi-cant and the lumped capacitance model applies. For a sphere this results in

θ

θi

= exp(−3BiFo) (2.73)

Q

Qi

= 1− exp(−3BiFo) (2.74)

Transient Conduction from Isothermal Bodies

Transient conduction from arbitrary three dimensional bodies at temperature To inan infinite medium of thermal conductivity k, thermal diffusivity α, and temperatureTi, may be accurately modelled using the Yovanovich, Teertstra, and Culham (1995)model:

Q? = 2√

π +1

√π√

Fo√A

(2.75)


where Q? =Q√

A

kA(To − Ti)and Fo√A =

αt

A. The length scale used to define the Fourier

number and dimensionless heat transfer rate is the square root of the body surfacearea.

2.5.2 Convection

Convective heat transfer models for internal and external flows are required for mod-elling heat exchangers, heat sinks, electronic enclosures, etc. A number of usefuldesign models and correlations are now presented for internal and external flows.

Internal Forced Convection

Circular Duct

For laminar developing flow in circular tubes with constant wall temperature, onemay use the following model due to Stephan. This correlation is valid for all valuesof the dimensionless duct length z∗ = L/(DReD) and for 0.1 < Pr < ∞:

Num,T =Nu(Pr →∞)

tanh(2.432Pr1/6(z∗)1/6)(2.76)

where

Nu(Pr →∞) =3.657

tanh(2.264(z∗)1/3 + 1.7(z∗)2/3)+

0.0499

z∗tanh(z∗) (2.77)

Plane Channel

For laminar developing flow in plane channels with constant wall temperature,one may use the following model, also due to Stephan, which is valid for all z∗ =L/(2bRe2b) and for 0.1 < Pr < 1000:

Num,T = 7.55 +0.024(z∗)−1.14

1 + 0.0358Pr0.17(z∗)−0.64(2.78)

General Non-Circular Ducts

In laminar flow, Muzychka and Yovanovich (2001) proposed the following modelfor developing laminar flows i n most circular and non-circular ducts:

Nu√A(z∗) =

{C1C2

(fRe√A

z∗

) 13

}5

+

{C3

(fRe√A

8√

πεγ

)}5)m/5

+

{C4f(Pr)√

z∗

}m1/m

(2.79)

Fundamentals 31

wherem = 2.27 + 1.65Pr1/3 (2.80)

andz∗ =

z√ARe√APr

(2.81)

and

fRe√A =12

ε1/2(1 + ε)

[1− 192ε

π5tanh

( π

2ε

)] (2.82)

Table 3Coefficients for General ModelLaminar Internal Flow Model

Boundary Condition

Isothermal C2 = 0.409, C3 = 3.24 f(Pr) =0.564[

1 + (1.664Pr1/6)9/2]2/9

Isoflux C2 = 0.501, C3 = 3.86 f(Pr) =0.886[

1 + (1.909Pr1/6)9/2]2/9

Nusselt Type

Local C1 = 1 C4 = 1

Average C1 = 3/2 C4 = 2

Shape Parameter

Symmetric Ducts γ = 1/10

Asymmetric Ducts γ = −3/10

In the above model, the characteristic length scale is the square root of the cross-sectional duct area. The parameter γ is chosen based upon the duct geometry. Thelower bound value is for ducts that have re-entrant corners, i.e. angles less than 90degrees. The upper bound is for ducts with rounded corners, rectangular or ellipticalshapes. The coefficients are tabulated in Table 3 for various conditions.

Turbulent Flows in Circular and Non-Circular Ducts

For turbulent flows the most popular expression is the correlation developed byGneilinski (1976).


NuDh=

(f/8)ReDhPr

1.07 + 12.7(f/8)1/2(Pr2/3 − 1)(2.83)

where

f = (0.79 ln Redh− 1.64)−2 (2.84)

External Forced Convection

Flate PlateFor a flat plate in laminar boundary layer flow, 1000 < ReL < 500, 000, the Nusselt

number is obtained from the following expressions:

Nux = (RexPr)1/2f(Pr) (2.85)

NuL = 2(ReLPr)1/2f(Pr) (2.86)

where for the constant surface temperature, Ts, boundary condition

f(Pr) =0.564[

1 + (1.664Pr1/6)9/2]2/9

(2.87)

and for the constant heat flux, qs, boundary condition

f(Pr) =0.886[

1 + (1.909Pr1/6)9/2]2/9

(2.88)

In turbulent boundary layer flow, 500, 000 < ReL < 107, the following equationsare often used:

Nux = 0.0296Re4/5x Pr1/3 (2.89)

NuL = 0.037Re4/5L Pr1/3 (2.90)

For a combined laminar/turbulent boundary layer, ReL > 500, 000, the followingintegrated expression is useful:

NuL = (0.037Re4/5L − 871)Pr1/3 (2.91)

Cylinder PeD > 0.2

For a cylinder in crossflow Churchill and Bernstein (1977) proposed the followingcorrelation of experimental data:

Fundamentals 33

NuD = 0.3 +0.62Re

1/2D Pr1/3

[1 + (0.4/Pr)2/3]1/4

[1 +

(ReD

282, 000

)5/8]4/5

(2.92)

Spheroids 0 < Re√A < 2× 105 and Pr > 0.7

For a sphere or spheroidal shaped body Yovanovich (1988) recommends the follow-ing model

Nu√A = 2√

π +

[0.15

(P√A

)1/2

Re1/2√

A+ 0.35Re0.566√

A

]Pr1/3 (2.93)

where A is the surface area and P is the maximum equitorial perimeter.

Internal Natural Convection

Parallel Plates

The Nusselt number for laminar natural convection flow between parallel isother-mal plates is obtained from the following correlation developed by Bar-Cohen andRohsenow (1984)

Nub =

[576

[Rab(b/L)]2+

2.87

[Rab(b/L)]1/2

]−1/2

(2.94)

The Nusselt number for laminar natural convection flow between parallel isofluxplates is obtained from the follow correlation developed by Bar-Cohen and Rohsenow(1984)

Nub =

[48

[Ra∗b(b/L)]2+

2.51

[Ra∗b(b/L)]2/5

]−1/2

(2.95)

where Rab = gβ∆Tb3/(αν) and Ra∗b = gβq′′b4/(kαν), and b is the plate spacing.

Circular and Non-Circular Ducts

For laminar natural convection of a gas (Pr ≈ 0.71) in vertical isothermal ducts,Yovanovich et al.(2001) recommend:

Nu√A =

2

Ra√A

(√A/L

)fRe√A

(√A

P

)2−n

+

0.6

(Ra√A

√A

L

)1/4−n

−1/n

(2.96)where


n =1.2

ε1/9(2.97)

and

fRe√A =12

ε1/2(1 + ε)

[1− 192ε

π5tanh

( π

2ε

)] (2.98)

In the above model, the characteristic length scale is the square root of the cross-sectional duct area.

External Natural Convection

Flate Plate

For a vertical isothermal wall the following correlation is recommended for laminarflow GrL < 109:

Nux = 0.503Ra1/4x f(Pr) (2.99)

NuL =4

3Ra

1/4L f(Pr) (2.100)

where

f(Pr) =

(Pr

(Pr + 0.986Pr1/2 + 0.492)

)1/4

(2.101)

A correlation which is valid for both the laminar and turbulent regions 10−1 <RaL < 1012 was proposed by Churchill and Chu (1975). Their correlation takes thefollowing form:

NuL =

(0.825 +

0.387Ra1/6L

[1 + (0.492/Pr)9/16]8/27

)2

(2.102)

Horizontal Cylinder

A correlation which is valid for both the laminar and turbulent regions 10−5 <RaL < 1012 was proposed by Churchill and Chu (1975). Their correlation takes thefollowing form:

NuD =

(0.60 +

0.387Ra1/6L

[1 + (0.559/Pr)9/16]8/27

)2

(2.103)

Sphere

For a sphere with Ra < 1011, the following correlation is recommended:

Fundamentals 35

NuD = 2 +0.589Ra

1/4D

[1 + (0.469/Pr)9/16]4/9(2.104)

Other Three Dimensional Bodies

For three dimensional bodies in any orientation, Yovanovich (1987) recommendsthe following correlation for 0 < Ra√A < 108:

Nu√A = 2√

π + Ra1/4√

Af(Pr) (2.105)

where

f(Pr) =0.67

[1 + (0.492/Pr)9/16]4/9(2.106)

and A is the surface area of the body.

2.5.3 Radiation

Radiative heat transfer transfer is determined using the Stefan-Boltzmann law:

q1−2 = εF1−2σ(T 41 − T 4

2 ) (2.107)

where ε is the surface emissivity, F1−2 is the view factor, and σ = 5.670e−8 W/(m2 ·K4), the Stefan-Boltzmann constant.

A number of common two surface enclosure problems are:

Parallel Plates

q1−2 =σ(T 4

1 − T 42 )

1

ε1

+1

ε2

− 1(2.108)

Concentric Cylinders

q1−2 =σ(T 4

1 − T 42 )

1

ε1

+1− ε2

ε2

(r1

r2

) (2.109)

Concentric Spheres

q1−2 =σ(T 4

1 − T 42 )

1

ε1

+1− ε2

ε2

(r1

r2

)2 (2.110)


Additional enclosure problems are discussed in all basic heat transfer texts. Formore information on radiative exchange and radiative properties, the student shouldrefer to the course text on heat transfer.

Fundamentals 37

References

Bar-Cohen, A. and Rohsenow, W.M., “Thermally Optimum Spacing of Vertical

Natural Convection Cooled Parallel Plates”, Journal of Heat Transfer, Vol. 106, 1984.

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Bejan, A., Advanced Engineering Thermodynamics, 1997, Wiley, New York, NY.

Bejan, A., G. Tsatsaronis, and Moran, M., Thermal Design and Optimization,

1996, Wiley, New York, NY.

Blevins, R.D., Applied Fluid Dynamics Handbook, 1984, Van Nostrand Reinhold,

New York, NY.

Churchill, S.W. and Chu, H.H.S., “Correlating Equations for Laminar and Tur-

bulent Free Convection from a Horizontal Cylinder”, International Journal of Heat

and Mass Transfer, Vol. 18, 1975, pp. 1323-1329.

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Moran, M.J. and Shapiro, H.N., Fundamentals of Engineering Thermodynamics,


Munson, B.R., Young, D.F., Okiishi, T.H., Fundamentals of Fluid Mechanics,


Muzychka, Y.S. and Yovanovich, M.M., “Forced Convection Heat Transfer in

the Combined Entry Region of Non-Circular Ducts”, Submitted to the 2001 Interna-

tional Mechanical Engineering Congress and Exposition, New York, NY, November,

2001.

Rohsenow, W.M., Hartnett, J.P., and Ganic, E., Handbook of Heat Transfer,

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Yovanovich, M.M., “General Expression for Forced Convection Heat and Mass

Transfer from Isopotential Spheroids”, AIAA Paper 88-0743, AIAA 26th Aerospace

Sciences Meeting and Exhibit, Reno, NV, January 11-14, 1988.

Yovanovich, M.M., “On the Effect of Shape, Aspect Ratio, and Orientation Upon

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82, 1987, pp. 121-129.

Yovanovich, M.M., Teertstra, P.M., and Culham, J.R. “Modelling Transient

Conduction From Isothermal Convex Bodies of Arbitrary Shape,” Journal of Ther-

mophysics and Heat Transfer, Vol. 9, 1995, pp. 385-390.

Yovanovich, M.M., Teertstra, P.M., and Muzychka, Y.S. “Natural Convec-

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