Chapter 2
Thermodynamics, Fluid Dynamics,
and Heat Transfer
2.1
Introduction
In this chapter we will review fundamental concepts from Thermodynamics, Fluid
Dynamics, and Heat Transfer. Each section first begins with a review of the fundamentals. Subsequently, a review of important equations and solutions to fundamental
problems from each of the three fields. This chapter is only intended to provide the
necessary reference material for the course. It is not intended as a substitute for the
basic texts used in the thermo-fluids courses. During this course extensive reference
will be made to the following texts dealing with thermo-fluid fundamentals. These
are:
1) Fundamentals of Fluid Mechanics, Potter and Wiggert
2) Fundamentals of Engineering Thermodynamics, Moran and Shapiro
3) Fundamentals of Heat and Mass Transfer, Incropera and DeWitt
Where possible, the use of robust design models or correlations which span a wide
range of flow conditions will be encouraged. These comprehensive models allow for
greater flexibility in the design and optimization of thermal systems. Whereas piecewise models, i.e. those which consider each different flow region separately, tend to
detract from the integrated design approach developed in the notes.
13
14
2.2
2.2.1
Mechanical Equipment and Systems
Thermodynamics
First Law of Thermodynamics
The First Law of Thermodynamics, better known as the conservation of energy will
be utilized for both open and closed systems throughout this course. We shall begin
by examining the different ways of stating the First Law for both open and closed
systems.
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
to the net heat input and the net work done on the system. In terms of instantaneous
transfer rates, the First Law may be written on a per unit time basis
dE
= Q̇ + Ẇ
(2.2)
dt
The First Law of Thermodynamics may also be written for an open system containing a number of inlets and outlets
µ
¶
¶
µ
Vi2
Ve2
dEcv
= ṁi ui + pi vi +
+ gzi − ṁe ue + pe ve +
+ gze + Q̇cv + Ẇcv (2.3)
dt
2
2
This equation states that the accumulation of energy within the control volume
must equal the net inflow of energy into the control volume minus the net outflow
of energy from the control volume plus the increase in energy due to work and heat
transfers. The sign convention adopted in these notes is that any work done on
a system is considered positive, while any work done by the system is considered
negative. This convention reflects the notion that work done on a system increases
the energy of the system, i.e. the use of a pump or compressor. Contemporary texts
in Thermodynamics have preferred the use of the “heat engine” convention which
reflects that useful work done by the system is considered positive. Either convention
may 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 thermodynamic
processes. A reversible process is one in which there is no production of entropy. For
a closed system the second law of thermodynamics states that
Z 2
δQ
≤ S2 − S1
(2.4)
T
1
We define the entropy production Sgen as the difference between the entropy change
and the entropy transfer such that
15
Fundamentals
Sgen = (S2 − S1 ) −
Z
1
2
δQ
≥0
T
(2.5)
The Second Law of Thermodynamics for an open system containing a number of
inlets and outlets becomes
X
dScv X Qj X
=
+
m i si −
me se + Ṡgen
dt
Tj
(2.6)
This equation states that the rate of entropy accumulation within the control volume is balanced by the net transfer of entropy through heat exchanges with the
surroundings plus the net flow of entropy into the control volume and the rate of
entropy production within the control volume. It should be noted that for a steady
state 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 a
steady state process.
2.2.3
Exergy
The first and second laws of thermodynamics may be combined to develop a new
relation which governs a new quantity Exergy. Exergy is a measure of the potential
of a thermodynamic system to do work. Unlike energy, exergy can be destroyed.
Exergy analysis, sometimes called availability analysis, is used quite frequently in the
design 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, S
is entropy, and V is the volume of the system. The reference or dead state as it is
referred is denoted by the subscript (·)o . We may also define exergy as an intensive
property, 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 )
For a closed system the exergy balance yields
¶
Z 2µ
To
1−
E2 − E1 =
δQ − [W − po (V2 − V1 )] − Ed
Tb
1
(2.9)
(2.10)
The term Ed = To Sgen , is the exergy which is destroyed due to irreversibilities in the
system. For an open system with a number of inlets and outlets the exergy balance
yields:
16
Mechanical Equipment and Systems
¶ X
¶
µ
µ
X
dEcv X
To
dVcv
+
ṁi ei −
ṁe ee − Ėd
1−
Q̇j − Ẇcv − po
=
dt
Tj
dt
(2.11)
where
e = (h − ho ) − To (s − so ) + V 2 /2 + gz
is the flow exergy.
Table 1
Dimensionless Groups
Group
Definition
hL
ks
ρV L
Reynolds Number
Re ∼
µ
ν
Prandtl Number
Pr ∼
α
VL
∼ ReP r
Peclet Number
Pe ∼
α
gβ∆T L3
Grashof Number
Gr ∼
ν2
gβ∆T L3
Rayleigh Number
Ra ∼ ∼
∼ GrP r
αν
hL
(q/A)L
∼
Nusselt Number
Nu ∼
kf ∆T
kf
Nu
Stanton Number
St ∼
ReP r
Nu
Colburn Factor
j∼
ReP r1/3
τ
Friction Coefficient
Cf ∼ 1 2
ρV
2
(∆p/L)(A/P )
Fanning Friction Factor
f∼
1
ρV 2
2
Biot Number
Bi ∼
(2.12)
17
Fundamentals
2.3
Dimensionless Groups
Before proceeding to the review of fluid dynamics and heat transfer models, a brief
discussion on the use of dimensionless quantities is required. A number of important
dimensionless quantities appear throughout the text. The student should familiarize
himself or herself with these parameters and their use. Table 1 summarizes the most
important groups that will be encountered during this course.
2.4
2.4.1
Fluid Dynamics
Conservation Equations
Conservation of mass and momentum for a control volume will be applied throughout
the course. Here we will merely state the general form as previously discussed in fluid
mechanics courses.
Conservation of Mass
X
X
dmCV
=
ṁi −
ṁe
dt
Conservation of Momentum
X
X
X
~ =
~ e (ρV
~ e Ae ) −
~ i (ρV
~ i Ai )
F
V
V
(2.13)
(2.14)
In addition, we will also apply Bernoulli’s equation for a number of incompressible
flows.
Bernoulli’s Equation
P2 V22
P1 V12
+
+ z1 =
+
+ z2 + hL
γ
2g
γ
2g
2.4.2
(2.15)
Internal Flows
When analyzing flow in ducting or piping systems as well as flow through mechanical
equipment, a number of design models and correlations are required for relating the
mass flow rate to the pressure drop of the working fluid. The most common method
is through the definition of the friction factor. The Fanning friction factor will be
adopted for this course. It is defined as follows:
A ∆p
Dh ∆p
τ
f = 1 2 = P1 L2 = 41 L2
ρu
ρu
ρu
2
2
2
(2.16)
18
Mechanical Equipment and Systems
where
4A
(2.17)
P
where A is the cross-sectional area and P is the perimeter of the duct. In fully
developed laminar flows the friction factor takes the following form:
Dh =
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 following
formula
fapp ReDh =
"µ
3.44
√
L∗
where
L∗ =
¶2
+ (f ReDh )2
#1/2
L
Dh ReDh
(2.19)
(2.20)
In circular tubes the flow is developing in a region where L∗ < 0.058. The entrance
length for flow development is
Le = 0.058DReD
Table 2
Typical values of f ReDh = C for
Non-Circular Ducts
Shape
f ReDh = C
Equilateral Triangle
13.33
Square
14.23
Pentagon
14.74
Hexagon
15.05
Octagon
15.41
Circle
16
Elliptic 2:1
16.82
Elliptic 4:1
18.24
Elliptic 8:1
19.15
Rectangular 2:1
15.55
Rectangular 4:1
18.23
Rectangular 8:1
20.58
Parallel Plates
24
(2.21)
19
Fundamentals
For turbulent flows the friction factor is predicted using the Colebrook relation.
This correlation is the basis for the Moody diagram
µ
¶
1
2.51
ǫ/D
√ = −2 log
√
+
(2.22)
3.7
fd
ReD fd
where the subscript d denotes the Darcy friction factor defined as:
∆p
fd = 1 L2
ρu
2
D
(2.23)
The entrance length for turbulent flow in a tube is
Le = 4.4D(ReD )1/6
(2.24)
In non-circular ducts we use the concept of the hydraulic diameter D = Dh = 4A/P
to compute an equivalent duct diameter.
2.4.3
External Flows
A number of important design equations for external fluid flows are required to relate
the free stream velocity to the overall drag force. The three most common geometries
are the flat plate, the cylinder, and the sphere.
Flat Plate
For laminar boundary layer flows, 1000 < ReL < 500, 000, the important parameters are the boundary layer thickness and the friction coefficient:
δ(x) =
Cf,x =
Cf =
5x
1/2
(2.25)
1/2
(2.26)
Rex
0.664
Rex
1.328
1/2
ReL
(2.27)
For turbulent boundary layer flows, 500, 000 < ReL < 107 , the boundary layer
thickness and friction coefficient are:
δ(x) =
Cf,x =
Cf =
0.38x
1/5
(2.28)
1/5
(2.29)
Rex
0.059
Rex
0.074
1/5
ReL
(2.30)
20
Mechanical Equipment and Systems
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
−
1/5
ReL
1742
ReL
(2.31)
Finally, a number of useful models for predicting drag on flat plates, cylinders, and
spheres in low Reynolds number flows are also provided. These models will provide
the building blocks for analysing a fluid component or system.
Flat Plate 0.01 < ReL < 500, 000
Cf =
2.66
7/8
ReL
+
1.328
1/2
ReL
(2.32)
Cylinder 0.1 < ReD < 250, 000
CD =
10
2/3
ReD
+ 1.0
(2.33)
Sphere 0.01 < ReD < 250, 000
CD =
6
24
+
+ 0.4
ReD 1 + Re1/2
D
(2.34)
where
CD , Cf =
F/A
1
ρu2
2
(2.35)
Note care must be taken to ensure the correct characteristic area A is chosen based
upon the geometry.
2.5
2.5.1
Heat Transfer
Conduction
1-Dimensional Steady Conduction
Steady one-dimensional conduction in plane walls, cylinders, and spheres is easily
analyzed using the resistance concept. The thermal resistance is defined such that
∆T = QRt
(2.36)
For a multi-component system containing j layers, the following thermal resistance
results are useful.
21
Fundamentals
Plane Wall
Rt =
X tj
1
1
+
+
hi A
kj A ho A
(2.37)
Cylinder
Rt =
X ln(roj /rij )
1
1
+
+
(2πri L)hi
(2πkj L)
(2πro L)ho
(2.38)
Sphere
X 1
1
+
Rt =
(4πri2 )hi
4πkj
µ
1
1
−
rij
roj
¶
+
1
(4πro2 )ho
(2.39)
Multi-Dimensional Steady Conduction
In two or three dimensions, heat transfer by means of conduction is best analyzed
using shape factors. Many multi-dimensional solutions of practical interest have been
obtained and are outlined below. The conduction shape factor S, is defined such that:
1
(2.40)
Sk
where R is the thermal resistance and k is the thermal conductivity of the medium.
The shape factor S, is only a function of the geometry of the system. The overall
heat transfer rate is then related to an appropriate temperature difference:
R=
Q = Sk∆T
(2.41)
where ∆T is the temperature difference between two isothermal surfaces.
A number of useful shape factors are tabulated in the handout.
Transient Conduction
Transient conduction in finite and semi-infinite regions are also of interest. The
following solutions are useful for modelling a number of thermal systems.
Semi-Infinite Regions
Isothermal Wall
T (x, t) − Ts
= erf
Ti − Ts
µ
x
√
2 αt
¶
(2.42)
22
Mechanical Equipment and Systems
qs (t) =
k(Ts − Ti )
√
παt
(2.43)
Isoflux Wall
T (x, t) − Ti =
2qs
p
αt/π
exp
k
µ
−x2
4αt
2qs
Ts (t) − Ti =
k
µ
¶
αt
π
qs x
−
erf c
k
¶1/2
µ
x
√
2 αt
¶
(2.44)
(2.45)
Surface Convection
√ ¶¸
¶¸ ·
µ
·
µ
x
h αt
hx h2 αt
erf c √ +
+ 2
− exp
k
k
k
2 αt
(2.46)
µ √ ¶
µ 2 ¶
h αt
h αt
Ts (t) − Ti
erf c
= 1 − exp
(2.47)
2
T∞ − Ti
k
k
√
µ √ ¶
µ 2 ¶
h αt
qs (t) αt
h αt
erf
c
= exp
(2.48)
k(T∞ − T i)
k2
k
T (x, t) − Ti
= erf c
T∞ − Ti
µ
x
√
2 αt
¶
Finite Regions
Transient conduction from finite one dimensional and multi-dimensional regions
may be analyzed using the following solutions. In the solutions below θ = T − Tf ,
θi = Ti − Tf , and Qi = ρcp V (Ti − Tf ). The notation adopted in this section follows
that of Yovanovich (1999).
Plane Wall
∞
where
X
θ
=
An exp(−δn2 F o) cos(δn X)
θi
n=1
An =
4 sin(δn )
2δn + sin(2δn )
(2.49)
(2.50)
The eigenvalues δn are determined from
δn sin(δn ) = Bi cos(δn )
(2.51)
In the expressions above, F o = αt/L2 , X = x/L, and Bi = hL/k. The heat flow
at the surface of the wall is determined from
23
Fundamentals
¶
∞ µ
X
Q
2Bi2
=1−
exp(−δn2 F o)
2 (Bi2 + Bi + δ 2 )
Qi
δ
n
n
n=1
(2.52)
Next if F o > 0.24, the series solutions for temperature and heat flow reduce to
single term approximations
where
θ
= A1 exp(−δ12 F o) cos(δ1 X)
θi
µ
¶
Q
2Bi2
=1−
exp(−δ12 F o)
2
2
2
Qi
δ1 (Bi + Bi + δ1 )
δ1 =
1.5708
√
[1 + (1.5708/ Bi)2.139 ]0.4675
(2.53)
(2.54)
(2.55)
Finally, if the Biot number is small (Bi < 0.2), spatial effects are no longer significant and the lumped capacitance model applies. For a plane wall this results
in
θ
= exp(−BiF o)
θi
(2.56)
Q
= 1 − exp(−BiF o)
Qi
(2.57)
Infinite Cylinder
∞
where
X
θ
=
An exp(−δn2 F o)J0 (δn R)
θi
n=1
An =
2J1 (δn )
2
δn (J0 (δn ) + J12 (δn ))
(2.58)
(2.59)
The eigenvalues δn are determined from
δn J1 (δn ) = J0 (δn )Bi
(2.60)
In the expressions above, F o = αt/a2 , R = r/a, and Bi = ha/k. The heat flow at
the surface of the cylinder is determined from
¶
∞ µ
X
Q
4Bi2
exp(−δn2 F o)
=1−
2 (Bi2 + δ 2 )
Qi
δ
n
n
n=1
(2.61)
Next if F o > 0.21, the series solutions for temperature and heat flow reduce to
single term approximations
24
Mechanical Equipment and Systems
θ
= A1 exp(−δ12 F o)J0 (δ1 R)
θi
¶
µ
Q
4Bi2
exp(−δ12 F o)
=1−
2
2
2
Qi
δ1 (Bi + δ1 )
(2.62)
(2.63)
where
δ1 =
2.4048
√
[1 + (2.4048/ 2Bi)2.238 ]0.4468
(2.64)
Finally, if the Biot number is small (Bi < 0.2), spatial effects are no longer significant and the lumped capacitance model applies. For an infinite cylinder this results
in
θ
= exp(−2BiF o)
θi
(2.65)
Q
= 1 − exp(−2BiF o)
Qi
(2.66)
Sphere
∞
where
X
θ
sin(δn R)
=
An exp(−δn2 F o)
θi
δn R
n=1
(2.67)
4[sin(δn ) − δn cos(δn )]
2δn − sin(2δn )
(2.68)
δn cos(δn ) = (1 − Bi) sin(δn )
(2.69)
An =
The eigenvalues δn are determined from
In the expressions above, F o = αt/a2 , R = a/L, and Bi = ha/k. The heat flow at
the surface of the sphere is determined from
¶
∞ µ
X
Q
6Bi2
=1−
exp(−δn2 F o)
2
2
2
Qi
δn (Bi − Bi + δn )
n=1
(2.70)
Next if F o > 0.18, the series solutions for temperature and heat flow reduce to
single term approximations
sin(δ1 R)
θ
= A1 exp(−δ12 F o)
θi
δ1 R
µ
¶
Q
6Bi2
=1−
exp(−δ12 F o)
Qi
δ12 (Bi2 − Bi + δ12 )
(2.71)
(2.72)
25
Fundamentals
where
δ1 =
3.14159
√
[1 + (3.14159/ 3Bi)2.314 ]0.4322
(2.73)
Finally, if the Biot number is small (Bi < 0.2), spatial effects are no longer significant and the lumped capacitance model applies. For a sphere this results in
2.5.2
θ
= exp(−3BiF o)
θi
(2.74)
Q
= 1 − exp(−3BiF o)
Qi
(2.75)
Convection
Convective heat transfer models for internal and external flows are required for modelling heat exchangers, heat sinks, electronic enclosures, etc. A number of useful
design models and correlations are now presented for internal and external flows.
Internal Forced Convection
Circular and Non-Circular Ducts
In laminar flow, Muzychka and Yovanovich (2001) proposed the following model
for developing laminar flows:
(
N u√A (z ∗ ) =  C1 C2
where
µ
f Re√A
z∗
¶ 13 )5
+

¾m 1/m
½ µ
¶¾5 !m/5 ½
√
f Re A
C4 f (P r)

√
√ γ
C3
+
8 πǫ
z∗
m = 2.27 + 1.65P r1/3
and
z∗ = √
z
ARe√A P r
(2.76)
(2.77)
(2.78)
and
f Re√A =
12
³ π ´¸
192ǫ
1/2
ǫ (1 + ǫ) 1 − 5 tanh
π
2ǫ
·
(2.79)
26
Mechanical Equipment and Systems
In the above model, the characteristic length scale is the square root of the crosssectional duct area. The parameter γ is chosen based upon the duct geometry. The
lower bound value is for ducts that have re-entrant corners, i.e. angles less than 90
degrees. The upper bound is for ducts with rounded corners, rectangular or elliptical
shapes. The coefficients are tabulated in Table 3 for various conditions.
For turbulent flows the most popular expression is the correlation developed by
Gneilinski (1976).
N uDh =
where
(f /8)ReDh P r
1.07 + 12.7(f /8)1/2 (P r2/3 − 1)
(2.80)
f = (0.79 ln Redh − 1.64)−2
(2.81)
Table 3
Coefficients for General Model
Boundary Condition
Isothermal
Isoflux
C2 = 0.409, C3 = 3.24
C2 = 0.501, C3 = 3.86
f (P r) = h
f (P r) = h
0.564
1+
9/2
(1.664P r1/6 )
0.886
9/2
1 + (1.909P r1/6 )
Nusselt Type
Local
C1 = 1
C4 = 1
Average
C1 = 3/2
C4 = 2
i2/9
i2/9
Shape Parameter
Upper Bound
γ = 1/10
Lower Bound
γ = −3/10
External Forced Convection
Flate Plate
For a flat plate in laminar boundary layer flow, 1000 < ReL < 500, 000, the Nusselt
number is obtained from the following expressions:
N ux = (Rex P r)1/2 f (P r)
(2.82)
27
Fundamentals
N uL = 2(ReL P r)1/2 f (P r)
(2.83)
where for the constant surface temperature, Ts , boundary condition
f (P r) = h
0.564
1+
9/2
(1.664P r1/6 )
and for the constant heat flux, qs , boundary condition
f (P r) = h
i2/9
0.886
1+
9/2
(1.909P r1/6 )
i2/9
(2.84)
(2.85)
In turbulent boundary layer flow, 500, 000 < ReL < 107 , the following equations
are often used:
1/3
N ux = 0.0296Re4/5
x Pr
4/5
N uL = 0.037ReL P r1/3
(2.86)
(2.87)
For a combined laminar/turbulent boundary layer, ReL > 500, 000, the following
integrated expression is useful:
4/5
N uL = (0.037ReL − 871)P r1/3
(2.88)
Cylinder P eD > 0.2
For a cylinder in crossflow Churchill and Bernstein (1977) proposed the following
correlation of experimental data:
"
¶5/8 #4/5
µ
1/2
0.62ReD P r1/3
ReD
N uD = 0.3 +
1+
[1 + (0.4/P r)2/3 ]1/4
282, 000
(2.89)
Spheroids 0 < Re√A < 2 × 105 and P r > 0.7
For a sphere or spheroidal shaped body Yovanovich (1988) recommends the following model
N u√A
#
"
µ
¶1/2
√
P
1/2
√
= 2 π + 0.15 √
P r1/3
Re√A + 0.35Re0.566
A
A
where A is the surface area and P is the maximum equitorial perimeter.
(2.90)
28
Mechanical Equipment and Systems
Internal Natural Convection
Parallel Plates
The Nusselt number for laminar natural convection flow between parallel isothermal plates is obtained from the following correlation developed by Bar-Cohen and
Rohsenow (1984)
¸−1/2
2.87
576
+
(2.91)
N ub =
[Rab (b/L)]2 [Rab (b/L)]1/2
The Nusselt number for laminar natural convection flow between parallel isoflux
plates is obtained from the follow correlation developed by Bar-Cohen and Rohsenow
(1984)
·
·
2.51
48
+
N ub =
∗
∗
2
[Rab (b/L)]
[Rab (b/L)]2/5
¸−1/2
(2.92)
where Rab = gβ∆T b3 /(αν) and Ra∗b = gβq ′′ b4 /(kαν), and b is the plate spacing.
Circular and Non-Circular Ducts
For laminar natural convection in vertical isothermal ducts, Yovanovich et al.(2001)
recommend:
N u √A


³√
´Ã
!2 −n  Ã
!1/4 −n −1/n
√
√

 Ra√A

A/L
A 
A
√



2
=
+ 0.6 Ra A


f Re√A
P
L


(2.93)
where
n=
1.2
ǫ1/9
(2.94)
and
12
(2.95)
³ π ´¸
192ǫ
ǫ1/2 (1 + ǫ) 1 − 5 tanh
π
2ǫ
In the above model, the characteristic length scale is the square root of the crosssectional duct area.
f Re√A =
·
External Natural Convection
Flate Plate
For a vertical isothermal wall the following correlation is recommended for laminar
29
Fundamentals
flow GrL < 109 :
N ux = 0.503Ra1/4
x f (P r)
(2.96)
4 1/4
N uL = RaL f (P r)
3
(2.97)
where
µ
f (P r) =
Pr
(P r + 0.986P r1/2 + 0.492)
¶1/4
(2.98)
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 the
following form:
N uL =
Ã
1/6
0.387RaL
0.825 +
[1 + (0.492/P r)9/16 ]8/27
!2
(2.99)
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 the
following form:
N uD =
Ã
1/6
0.387RaL
0.60 +
[1 + (0.559/P r)9/16 ]8/27
!2
(2.100)
Sphere
For a sphere with Ra < 1011 , the following correlation is recommended:
1/4
N uD = 2 +
0.589RaD
[1 + (0.469/P r)9/16 ]4/9
(2.101)
Other Three Dimensional Bodies
For three dimensional bodies in any orientation, Yovanovich (1987) recommends
the following correlation for 0 < Ra√A < 108 :
√
1/4
N u√A = 2 π + Ra√A f (P r)
where
f (P r) =
0.67
[1 + (0.492/P r)9/16 ]4/9
and A is the surface area of the body.
(2.102)
(2.103)
30
2.5.3
Mechanical Equipment and Systems
Radiation
Radiative heat transfer transfer is determined using the Stefan-Boltzmann law:
q1−2 = ǫF1−2 σ(T14 − T24 )
(2.104)
where ǫ is the surface emissivity, F1−2 is the view factor, and σ = 5.670e−8 W/(m2 ·
K 4 ), the Stefan-Boltzmann constant.
A number of common two surface enclosure problems are:
Parallel Plates
q1−2 =
σ(T14 − T24 )
1
1
+ −1
ǫ1 ǫ2
(2.105)
Concentric Cylinders
q1−2
σ(T14 − T24 )
µ ¶
=
1
1 − ǫ 2 r1
+
ǫ1
ǫ2
r2
(2.106)
Concentric Spheres
q1−2 =
σ(T14 − T24 )
µ ¶2
1
1 − ǫ2 r1
+
ǫ1
ǫ2
r2
(2.107)
Additional enclosure problems are discussed in all basic heat transfer texts. For
more information on radiative exchange and radiative properties, the student should
refer to the course text on heat transfer.
Fundamentals
2.6
31
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.
Bejan, A. Heat Transfer, 1993, Wiley, New York.
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.
Churchill, S.W. and Chu, H.H.S., “Correlating Equations for Laminar and Turbulent Free Convection from a Horizontal Cylinder”, International Journal of Heat
and Mass Transfer, Vol. 18, 1975, pp. 1323-1329.
Churchill, S.W. and Bernstein, M., “A Correlating Equation for Forced Convection from gases and Liquids to a Circular Cylinder in Cross Flow”, Journal of Heat
Transfer, Vol. 99, 1977, pp. 300-306.
Churchill, S.W., “A Comprehensive Correlating Equation for Forced Convection
from Flat Plates”, American Institute of Chemical Engineers, Vol. 22, 1976, pp.
264-268.
Gnielinski, V., “New Equations for Heat and Mass Transfer in Turbulent Pipe and
Channel Flow”, International Chemical Engineering, Vol. 16, 1976, pp. 359-368.
Incropera, F.P. and DeWitt, D.P., Fundamentals of Heat and Mass Transfer,
1996, Wiley, New York, NY.
Moran, M.J. and Shapiro, H.N., Fundamentals of Engineering Thermodynamics,
2000, Wiley, New York, NY.
Munson, B.R., Young, D.F., Okiishi, T.H., Fundamentals of Fluid Mechanics,
1998, Wiley, New York, NY.
Muzychka, Y.S. and Yovanovich, M.M., “Forced Convection Heat Transfer in
the Combined Entry Region of Non-Circular Ducts”, Submitted to the 2001 International Mechanical Engineering Congress and Exposition, New York, NY, November,
2001.
Rohsenow, W.M., Hartnett, J.P., and Cho, Y.I., Handbook of Heat Transfer,
1999, McGraw-Hill, New York.
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.
32
Mechanical Equipment and Systems
Yovanovich, M.M., “On the Effect of Shape, Aspect Ratio, and Orientation Upon
natural Convection from Isothermal Bodies of Complex Shape”, ASME HTD Vol.
82, 1987, pp. 121-129.
Yovanovich, M.M., Teertstra, P.M., and Muzychka, Y.S. “Natural Convection Inside Vertical Isothermal Ducts of Constant Arbitrary Cross-Section”, AIAA
Paper 01-0368, AIAA 39th Aerospace Sciences Meeting and Exhibit, Reno, NV, January 8-11, 2001.