Heat Transfer Handbook part 23 pot

The differential equation governing the temperature in the ﬁn is with the boundary conditions dx where is the emissivity ofthe ﬁn surface and σ is the Stefan–Boltzmann constant σ = 5.6

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Concave parabolic Conical Constant cross section Convex parabolic

Figure 3.25 Efﬁciencies of convecting spines

3.6.4 Longitudinal Radiating Fins

Unlike convecting fins, for which exact analytical solutions abound, few such so-lutions are available for radiating fins Consider the longitudinal fin of rectangular

proﬁle shown in Fig 3.19a and let the ﬁn radiate to free space at 0 K The differential

equation governing the temperature in the ﬁn is

with the boundary conditions

dx

where is the emissivity ofthe ﬁn surface and σ is the Stefan–Boltzmann constant

(σ = 5.667 × 10−8W/m2· K4)

The solution for the temperature distribution, rate of heat transfer, and ﬁn efﬁ-ciency are

20σT3

t kδ

1/2

(3.214)

qf = 2kδL σ

5kδ

1/2

b − T5

t

1/2

(3.215)

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212 CONDUCTION HEAT TRANSFER 1

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b − T5

b

1/2

2σbLT4

b

(3.216)

whereB and B uare complete and incomplete beta functions discussed in Section 3.3.3,u = (T t /T )5andT tis the unknown tip temperature BecauseT tis not known, the solution involves a trial-and-error procedure

Sen and Trinh (1986) reported the solution ofeqs (3.212) and (3.213) when the surface heat dissipation is proportional toT mrather thanT4 Their solution appears

in terms ofhypergeometric functions which bear a relationship to the incomplete beta function Kraus et al (2001) provide an extensive collection of graphs to evaluate the performance ofradiating ﬁns ofdifferent proﬁles

3.6.5 Longitudinal Convecting–Radiating Fins

A finite-difference approach was taken by Nguyen and Aziz (1992) to evaluate the performance oflongitudinal fins (Fig 3.19) ofrectangular, trapezoidal, triangular, and concave parabolic profiles when the fin surface loses heat by simultaneous con-vection and radiation For each profile, the performance depends on five parameters,

b /kδb, whereTsis the effective sink tem-perature for radiation A sample result for the ﬁn efﬁciency is provided in Table 3.11

These results reveal a more general trend—that a convecting–radiating fin has a lower efficiency than that of a purely convecting fin(2b2σT3

b /kδ b = 0)

3.6.6 Optimum Dimensions of Convecting Fins and Spines

The classical fin or spine optimization involves finding the profile so that for a pre-scribed volume, the fin or spine rate ofheat transfer is maximized Such optimizations result in profiles with curved boundaries that are difficult and expensive to fabricate

From a practical point ofview, a better approach is to select the profile first and then find the optimum dimensions so that for a given profile area or volume, the fin or spine rate ofheat transfer is maximized The results ofthe latter approach are provided here

For each shape, two sets of expressions for optimum dimensions are given, one set for

TABLE 3.11 Efﬁciency of Longitudinal Convecting–Radiating Fins,

T∞/Tb = T s/Tb = 0.8, 2hb2/kδb= 1

2b2σT3

b /kδ Rectangular δt/δ b = 0.25 Triangular Parabolic

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when the profile area or volume is specified and another set for when the fin or spine rate ofheat transfer is specified Note thatq f for fins in the expressions to follow is the fin rate ofheat transfer per unit lengthL offin.

Rectangular Fin When the weight or proﬁle areaApis speciﬁed,

δopt = 0.9977

p h k

1/3

(3.217)

bopt = 1.0023

h

1/3

(3.218) and when the ﬁn rate ofheat transfer (per unit length)q f is speciﬁed,

Triangular Fin When the weight or proﬁle areaA p is speciﬁed,

δb,opt = 1.6710

p h k

1/3

(3.221)

bopt = 1.1969

A

p k h

1/3

(3.222) and when the ﬁn rate ofheat transfer (per unit length)qf is speciﬁed,

δb,opt =0.8273

Concave Parabolic Fin When the weight or proﬁle areaA pis speciﬁed,

δopt = 2.0801

p h k

1/3

(3.225)

bopt = 1.4422

A

p k h

1/3

(3.226) and when the ﬁn rate ofheat transfer (per unit length)q f is speciﬁed,

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δb,opt =

1

hk

qf

2

(3.227)

Cylindrical Spine When the weight or volumeV is speciﬁed,

dopt= 1.5031

k

1/5

(3.229)

bopt= 0.5636

V k2

1/5

(3.230) and when the spine rate ofheat transferqf is speciﬁed,

dopt = 0.9165

f

1/3

(3.231)

bopt = 0.4400

1/3

(3.232)

Conical Spine When the weight or volumeV is speciﬁed,

hV2

k

1/5

(3.233)

bopt = 1.0008

V k2

1/5

(3.234) and when the spine rate ofheat transferq f is speciﬁed,

d b,opt = 1.0988

f

1/3

(3.235)

bopt = 0.7505

q

f k

1/3

(3.236)

Concave Parabolic Spine When the weight or volumeV is speciﬁed,

d b,opt = 2.0968

k

1/5

(3.237)

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bopt= 1.4481

V k2

1/5

(3.238) and when the spine rate ofheat transferqf is speciﬁed,

d b,opt = 1.1746

f

1/3

(3.239)

bopt = 1.0838

1/3

(3.240)

Convex Parabolic Spine When the weight or volumeV is speciﬁed,

hV2

k

1/5

(3.241)

bopt= 0.7877

1/5

(3.242) and when the spine rate ofheat transferq f is speciﬁed,

f

1/3

(3.243)

bopt = 0.5951

q

f k

1/3

(3.244)

The material presented here is but a small fraction ofthe large body ofliterature

on the subject ofoptimum shapes ofextended surfaces The reader should consult Aziz (1992) for a comprehensive compilation of results for the optimum dimensions ofconvecting extended surfaces Another article by Aziz and Kraus (1996) provides similar coverage for radiating and convecting–radiating extended surfaces Both ar-ticles contain a number ofexamples illustrating the design calculations, and both are summarized in Kraus et al (2001)

3.7 TWO-DIMENSIONAL STEADY CONDUCTION

The temperature ﬁeld in a two-dimensional steady-state conﬁguration is controlled by

a second-order partial differential equation whose solution must satisfy four boundary conditions The analysis is quite complex, and consequently, exact analytical solu-tions are limited to simple geometries such as a rectangular plate, a cylinder, and a sphere under highly restrictive boundary conditions Problems that involve complex geometries and more realistic boundary conditions can only be solved by using an

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approximate technique or a numerical method Approximate techniques that are em-ployed include the integral method, the method ofscale analysis, and the method of conduction shape factors The two most popular numerical techniques are the ﬁnite-difference and ﬁnite-element methods There are numerous sources for information

on approximate and numerical techniques, some ofwhich are Bejan (1993), Ozisik (1993, 1994), Comini et al (1994), and Jaluria and Torrance (1986) In the follow-ing section we provide an example ofan exact solution, a table ofconduction shape factors, and a briefdiscussion ofthe ﬁnite-difference method and its application to two-dimensional conduction in a square plate and a solid cylinder

3.7.1 Rectangular Plate with Speciﬁed Boundary Temperatures

Figure 3.26 shows a rectangular plate with three sides maintained at a constant temperatureT1, while the fourth side is maintained at another constant temperature,

(3.245) the governing two-dimensional temperature distribution becomes

with the boundary conditions

θ(0,y) = 1 (3.247a)

θ(x,0) = 0 (3.247b)

θ(L,y) = 0 (3.247c)

θ(x,H ) = 0 (3.247d) Use ofthe separation ofvariables method gives the solution forθ as

θ = 4 π

∞

n=0

sinh [(2n + 1)π(L − x)/H ]

sinh [(2n + 1)πL/H ]

sin [(2n + 1)πy/H ]

Using eq (3.248), Bejan (1993) developed a network ofisotherms and heat ﬂux lines, which is shown in Fig 3.27 forH/L = 2 (a rectangular plate) and for H/L = 1 (a

square plate)

The heat ﬂow into the plate from a hot left face is given by

q

8

πk(T2− T1)

∞

n=0

1

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Figure 3.26 Two-dimensional steady conduction in a rectangular plate

0.2 0

0

H

L = 2

L = 1

Isotherm Heat flux line

Figure 3.27 Isotherms and heat ﬂux lines in a rectangular plate and a square plate (From Bejan, 1993.)

whereW is the plate dimension in the z direction Solutions for the heat ﬂux and

convective boundary conditions are given in Ozisik (1993) and Poulikakos (1994)

3.7.2 Solid Cylinder with Surface Convection

Figure 3.28 illustrates a solid cylinder ofradiusr0and lengthL in which conduction

occurs in both radial and axial directions The face atz = 0 is maintained at a constant

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Figure 3.28 Radial and axial conduction in a hollow cylinder

temperature T1, while both the lateral surface and the face atz = L lose heat by

convection to the environment atT∞via the heat transfer coefﬁcienth The system

described represents a two-dimensional (r, z) convecting spine discussed by Aziz and

Lunardini (1995) The equation governing the two-dimensional heat conduction in the cylinder is

∂R2 +R1 ∂R ∂θ +∂Z ∂2θ2 = 0 (3.250) where

L

and Bi is the Biot number, Bi = hr0/k The boundary conditions are

θ(R,0) = 1 (3.251a)

∂θ

The solution obtained via the separation ofthe variables is

θ =

∞

n=1

2λn J1(λ n )J0(λ n R)

λ2

n+ Bi2

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where

Υ = λλnsinhλnγ + Bi coshλnγ

ncoshλnγ + Bi sinhλnγ

and whereJ0andJ1are the Bessel functions of the ﬁrst kind (Section 3.3.5) and the eigenvaluesλnare given by

The heat ﬂow into the cylinder from the hot left face is

∞

n=1

λn[J1(λ n )]2

λ2

n+ Bi2

A three-dimensional plot ofθ as a function of r and z is shown in Fig 3.29 for

r0 = 1, L = 1, and h/k = 1 This plot was generated using Maple V, Release 5.0 As

expected, the temperature decreases along both the radial and axial directions Ozisik (1993) has devoted a complete chapter to the method ofseparation ofvariables in cylindrical coordinates and provides solutions for several other conﬁgurations

3.7.3 Solid Hemisphere with Speciﬁed Base and Surface Temperatures

Poulikakos (1994) considers a hemispherical droplet condensing on a cold horizontal surface as shown in Fig 3.30 The heat conduction equation for the two-dimensional

∂

∂r

+ 1

sinθ

∂

∂θ

sinθ∂φ

∂θ

whereφ = T − T c Two ofthe boundary conditions are

φr,π

2

Because the boundary condition atr = 0 falls on the θ = π/2 plane, which is

the base ofthe hemispherical droplet, it must meet the boundary condition ofeq

(3.256b), that is,

φr,π

2

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Figure 3.29 Three-dimensional plot ofthe temperature distribution in a solid cylinder (From Aziz, 2001.)

The fourth boundary condition atθ = 0 is obtained by invoking the condition of

thermal symmetry aboutθ = 0, giving

∂φ

Use ofthe method ofseparation ofthe variables provides the solution forφ as

φ = φs

∞

n=1

n

where theP ’s are the Legendre functions of the ﬁrst kind, discussed in Section 3.3.6.

Ozisik (1993) may be consulted for a comprehensive discussion of the method of separation ofthe variables in spherical coordinates

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