In this section we provide the performance characteristics temperature distribu-tion, rate of heat transfer, and fin efficiency for convecting, radiating, and convecting-radiating fins.. Th
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3.5.5 Radiative–Convective Cooling of Solids with Uniform Energy Generation
The solutions obtained in Section 3.4.8 for a plane wall (the thermal symmetry case), a solid cylinder, and a solid sphere are now extended to accommodate surface cooling by simultaneous convection and radiation The surface energy balance for each geometry gives
h(T s − T∞) + σT4
s − T4
∞
+ k dT dx
x=L= 0 (plane wall) (3.170)
h(T s − T∞) + σT4
s − T4
∞
+ k dT dr
r=r0= 0 (solid cylinder and sphere) (3.171)
where is the surface emissivity, σ the Stefan–Boltzmann constant, and T∞represents the surrounding or ambient temperature for both convection and radiation In eqs
(3.170) and (3.171), the last terms can be evaluated using eqs (3.105), (3.115), and (3.123), respectively
Because eqs (3.170) and (3.171) require a numerical approach for their solutions,
it is convenient to recast them in dimensionless form as
N1(θ s − 1) + N2θ4
s − 1+ dθ
dX
X=1= 0 (plane wall) (3.173)
N1(θ s − 1) + N2θ4
s− 1+ dθ
dR
R=1= 0 (cylinder and sphere) (3.173) whereθ = T s /T∞, N1 = hL/k for the plane wall, N1 = hr0/k for the cylinder and
sphere,N2 = σT3
∞L/k for the plane wall, and N2 = σT3
∞r0/k for the cylinder and
sphere,X = x/L, R = r/r0, andθ = T /T∞ The numerical values forθsare given
in Table 3.10 for a range of values ofN1 andN2 and ˙qL2/kT∞ = 1 for the plane
wall and ˙qr2
0/kT∞= 1 for the cylinder and sphere
The term extended surface is used to describe a system in which the area ofa surface
is increased by the attachment of fins A fin accommodates energy transfer by
conduc-tion within its boundaries, while its exposed surfaces transfer energy to the surround-ings by convection or radiation or both Fins are commonly used to augment heat transfer from electronic components, automobile radiators, engine and compressor cylinders, control devices, and a host ofother applications A comprehensive treat-ment ofextended surface technology is provided by Kraus et al (2001)
In this section we provide the performance characteristics (temperature distribu-tion, rate of heat transfer, and fin efficiency) for convecting, radiating, and convecting-radiating fins Configurations considered include longitudinal fins, radial fins, and spines The section concludes with a discussion ofoptimum fin designs
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Generation and Radiative–Convective Surface Cooling
θs
3.6.1 Longitudinal Convecting Fins
The five common profiles oflongitudinal fins shown in Fig 3.19 are rectangular, trapezoidal, triangular, concave parabolic, and convex parabolic The analytical ex-pressions obtained are based on several assumptions
1 The heat conduction in the fin is steady and one-dimensional
2 The fin material is homogeneous and isotropic
3 There is no energy generation in the fin
4 The convective environment is characterized by a uniform and constant heat transfer coefficient and temperature
5 The fin has a constant thermal conductivity
6 The contact between the base ofthe fin and the primary surface is perfect
7 The fin has a constant base temperature
Rectangular Fin For the rectangular fin (Fig 3.19a), the temperature distribution,
rate of heat transfer, and fin efficiency are given for five cases of thermal boundary conditions
1 Constant base temperature and convecting tip:
θ
θb =
coshm(b − x) + H sinhm(b − x)
coshmb + H sinhmb (3.174)
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q f = kmAθ b sinhmb + H coshmb
coshmb + H sinhmb (3.175)
η = q f
T ,h⬁
T ,h⬁
T ,h⬁
T ,h⬁
T ,h⬁
T ,h⬁
T ,h⬁
T ,h⬁
T ,h ⬁ t
T b
T b
T b
T b
T b
k
k
k
k
k b
b
b
b
b
L
L
L
L
L
x
x
x
x
x
q f
q f
q f
q f
q f
x e
␦b
␦b
␦b
A
( )a
( )c
( )e
( )b
( )d
P
= L
= fin perimeter 2( ⫹ δ)
T ,h⬁
T ,h⬁
insulated tip
parabolic, and (e) convex parabolic profiles.
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whereθ = T − T∞, θ b = T b − T∞, m2 = hP /kA = 2h/kδ, H = h t /km, and
T bis the fin base temperature,T the fin temperature at location x, T∞the convective
environmental temperature,b the fin height, A the fin cross-sectional area, P the fin
perimeter,k the fin thermal conductivity, h the convective heat transfer coefficient for
surfaces other than the fin tip,h t the tip convective heat transfer coefficient,q f the
fin heat dissipation, andq id the ideal fin heat dissipation.
2 Constant base temperature and insulated tip ( H = 0):
θ
θb =
coshm(b − x)
q f = kmAθ btanhmb (3.179)
η = tanhmb
3 Constant base and tip temperatures:
θ
θb = (θ t /θ b ) sinhmx + sinhm(b − x)
q f = kmAθ bcoshmb − (θ t /θ b )
withqid andη given by eqs (3.176) and (3.177), respectively, T t taken as the
pre-scribed tip temperature, andθt = T t − T∞
4 Convective heating at the base and insulated tip:
θ
θf =
Bi cosh(mb − x)
q f = kmAθ f Bi sinhmb
Bi coshmb + mb sinhmb (3.184)
where Bi= h f b/k, θ f = T f −T∞, andh fandT fcharacterize the convection process
at the fin base Equations (3.176) and (3.177) can be used to findqid andη, but θb
must be found first from eq (3.183)
5 Infinitely high fin with constant base temperature:
θ
Because the fin is infinitely high,qidandη cannot be calculated Instead, one may
calculate the fin effectiveness as the ratio of q f to the rate of heat transfer from the
base surface without the fin, hAθ b Thus
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= q f
hAθ b =
kP hA
1/2
(3.187) Several important conclusions can be drawn from eq (3.187) First, the fin effec-tiveness is enhanced by choosing a material with high thermal conductivity Copper has a high value (k = 401 W/m · K at 300K), but it is heavy and expensive Aluminum
alloys have lowerk (k = 168 to 237 W/m · K at 300 K) but are lighter, offer lower
cost, and in most instances are preferable to copper Second, the fins are more effective when the convecting fluid is a gas (lowh) rather than a liquid (higher h) Moreover,
there is a greater incentive to use the fin under natural convection (lowerh) than under
forced convection (higherh) Third, the greater the perimeter/area (P /A) ratio, the
higher the effectiveness This, in turn, suggests the use of thin, closely spaced fins
However, the gap between adjacent fins must be sufficient to prevent interference of the boundary layers on adjacent surfaces
Trapezoidal Fin For a constant base temperature and insulated tip, the tempera-ture distribution, rate of heat transfer, ideal rate of heat transfer, and fin efficiency for
a trapezoidal fin (Fig 3.19b) are
θ
θb =I0(2m
√
bx)K1(2m√bx e ) + K0(2m√bx)I I (2m√bx e ) I0(2mb)K1(2m√bx e ) + K0(2mb)I1(2m√bx e ) (3.188)
q f = kmδ b Lθ b I1(2mb)K1(2m√bx e ) − K1(2mb)I I (2m√bx e )
I0(2mb)K1(2m√bx e ) + K0(2mb)I1(2m√bx e ) (3.189)
and eq (3.177) gives the fin efficiency In eqs (3.188) and (3.189),m = √2h/kδ b
andx eis the distance to the fin tip The modified Bessel functions appearing here and
in subsequent sections are discussed in Section 3.3.5
Triangular Fin The rectangular fin (Fig 3.19c) is a special case ofthe trapezoidal
fin withx e= 0 and
θ
θb =
I0(2m√bx)
q f = kmδ b Lθ b I1(2mb)
η = mbI0(2mb) I1(2mb) (3.193)
Concave Parabolic Fin For the concave parabolic fin shown in Fig 2.19d, the
temperature distribution, rate ofheat transfer, and fin efficiency are
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θ
θb =
x b
−1/2+1/2(1+4m2b2)1/2
(3.194)
q f =kδ b Lθ b
2b
−1 +1+ 4m2b21/2
(3.195)
1+1+ 4m2b21/2 (3.196)
Convex Parabolic Fin For the convex parabolic fin shown in Fig 3.19e, the
temperature distribution, rate ofheat transfer, and fin efficiency are
θ
θb =
x b
1/4I −1/34
3mb1/4 x3/4
I −1/34
3mb
(3.197)
q f = kmδ b Lθ b I2 /34
3mb
I −1/34
η = 1
mb
I2 /34
3mb
I −1/34
The efficiency oflongitudinal fins ofrectangular, triangular, concave parabolic, and convex parabolic fins are plotted as a function ofmb in Fig 3.20.
3.6.2 Radial Convecting Fins
The radial fin is also referred to as an annular fin or circumferential fin, and the
performance of three radial fin profiles is considered These are the rectangular, triangular, and hyperbolic profiles Analytical results are presented for the rectangular profile, and graphical results are provided for all three profiles
Rectangular Fin For the radial fin ofrectangular profile shown in the inset of Fig 3.21, the expressions for the temperature distribution, rate of heat transfer, and
fin efficiency are
θ
θb =
K1(mr a )I0(mr) + I1(mr a )K0 (mr) I0(mr b )K1(mr a ) + I1(mr a )K0 (mr b ) (3.200)
q f = 2πr b kmδθ b I1(mr a )K1(mr b ) − K1(mr a )I1(mr b )
I0(mr b )K1(mr a ) + I1(mr a )K0(mr b ) (3.201)
η = 2r b
mr2
a − r2
b
I1(mr a )K1(mr b ) − K1(mr a )I1(mr b )
I0(mr b )K1(mr a ) + I1(mr a )K0(mr b ) (3.202)
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1.0 0.9 0.8 0.7 0.6 0.5 0.4
mb
Rectangular Convex parabolic Triangular Concave parabolic
1.0 0.9 0.8 0.7 0.6
0.3 0.5
0.2 0.4
0.1 0
Fin parameter, =m L公2 / h k ␦b
1 1.5 2 3 4 5
rb
␦b
ra b
r r a b/
and Kalman, 1989.)
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1.0 0.9 0.8 0.7 0.6
0.3 0.5
0.2 0.4
0.1 0
Fin parameter, =m b公2 / h k ␦b
1.5 2 3 5
rb
␦b
ra b
r r a b/
␦b b /r
0.01 0.6 0.01–0.6
and Kalman, 1989.)
The efficiency ofa radial fin ofrectangular profile given by eq (3.202) is plotted as
a function ofmb in the main body ofFig 3.21 for r a /r b = 1 (longitudinal fin), 1.5,
2.0, 3.0, 4.0, and 5.0
Triangular Fin The inset in Fig 3.22 shows a radial fin oftriangular profile The analysis for this profile is given in Kraus et al (2001) and involves an infinite series that is omitted in favor of numerical results for the fin efficiency, which are graphed
in Fig 3.22 Note thatη is a function of m, r a /r b, and δb /r b Onceη is known,
q f = 2π(r2
a − r2
b )hθ bη
Hyperbolic Fin A radial fin ofhyperbolic profile appears as an inset in Fig 3.23
The lengthy analytical results are presented in Kraus et al (2001) and a graph ofthe
fin efficiency is presented in Fig 3.23 Note thatη is a function of m, r a /r b, andδb /r b, and onceη is known, q f = 2π(r2
a − r2
b )hθ bη
3.6.3 Convecting Spines
Four commonly used shapes ofspines, shown in Fig 3.24, are the cylindrical, coni-cal, concave parabolic, and convex parabolic Analytical results for the temperature distribution, rate of heat transfer, and fin efficiency are furnished
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1.0 0.9 0.8 0.7 0.6
0.3 0.5
0.2 0.4
0.1 0
Fin parameter, =m b公2 / h k ␦b
1.5 2 3 4 5
rb
␦b
ra b
␦b b /r
0.01 0.6 0.01–0.6 1
and Kalman, 1989.)
Cylindrical Spine For the cylindrical spine, the results for the rectangular fins are applicable ifm = (4h/kd)1/2is used instead ofm = (2h/kδ)1/2 Ifthe spine tip is
insulated, eqs (3.178)–(3.180) can be used
Conical Spine
θ
θb =
b x
1/2
I1(2M√x)
q f =πkd b2Mθ b
4√
b
I2(2M√b)
η = 2
M√b)
I2(2M√b)
whereM = (4hb/kd b )1/2.
Concave Parabolic Spine
θ
θb =
x b
−3/2+1/2(9+4M2)1/2
(3.206)
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q f =πkd b2θb
−3 + (9 + 4M2)1/2
1+ (1 +8
whereM = (4hb/kd b )1/2andm = (2h/kd b )1/2.
Convex Parabolic Spine
θ
θb =I0
4
3Mx3/4
I04
q f =πkd b2Mθ b
2b1/4
I14
3Mb3/4
I04
η = 3
2√
2
I14 3
√
2mb mbI04 3
√
whereM = (4hb1/2 /kd b )1/2andm = (2h/kd b )1/2.
Figure 3.25 is a plot ofη as a function of mb for the four spines discussed.
T h a,
T h a,
T h a,
T h a,
T b
T b
T b
T b
k
k
k k
b
b
b
b
d b
d
d b
d b
( )c
( )a
( )d
( )b