in the radial direction with no internal heat generation and constant thermal conduc-tivity, the appropriate form of the general heat conduction equation, eq.. With this definition, the t
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Figure 3.7 Radial conduction through a hollow cylinder
in the radial direction with no internal heat generation and constant thermal conduc-tivity, the appropriate form of the general heat conduction equation, eq (3.5), is
d dr
r dT dr
with the boundary conditions expressed as
T (r = r1) = T s,1 and T (r = r2) = T s,2 (3.70) Following the same procedure as that used for the plane wall will give the temperature distribution
T = T s,1+T s,1 − T s,2
ln(r1/r2) ln
r
and the heat flow
q = 2πkL(T x,1 − T s,2 )
3.4.3 Hollow Sphere
The description pertaining to the hollow cylinder also applies to the hollow sphere ofFig 3.8 except that the lengthL is no longer relevant The applicable form of eq.
(3.6) is
d dr
r2dT dr
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Figure 3.8 Radial conduction through a hollow sphere
with the boundary conditions expressed as
T (r = r1) = T s,1 and T (r = r2) = T s,2 (3.74) The expressions for the temperature distribution and heat flow are
T = T s,1+ T s,1 − T s,2
1/r2 − 1/r1
1
r1 −
1
r
(3.75)
q = 4πk(T s,1 − T s,2 )
3.4.4 Thermal Resistance
Thermal resistance is defined as the ratio ofthe temperature difference to the
associ-ated rate ofheat transfer This is completely analogous to electrical resistance, which, according to Ohm’s law, is defined as the ratio ofthe voltage difference to the current flow With this definition, the thermal resistance ofthe plane wall, the hollow cylinder, and the hollow sphere are, respectively,
Rcond= ln(r2/r1)
Rcond= 1/r1 − 1/r2
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thermal network shown in Fig 3.9, the rate ofheat transferq is given by
q = T ∞,1 − T ∞,2
1/h1A + L1/k1A + L2/k2A + 1/h2A (3.82)
Onceq has been determined, the surface and interface temperatures can be found:
T s,1 = T ∞,1 − q 1
T2 = T ∞,1 − q
1
h1A+
L1 k1A
(3.84)
T s,2 = T ∞,1 − q
1
h1A+
L1 k1A+
L2 k2A
(3.85)
Figure 3.10 illustrates a series–parallel composite wall Ifmaterials 2 and 3 have comparable thermal conductivities, the heat flow may be assumed to be one-dimen-sional The network shown in Fig 3.10 assumes that surfaces normal to the direction ofheat flow are isothermal For a wall ofunit depth, the heat transfer rate is
L1/k1H1 + L2L3/(k2H2L3 + k3H3L2) + L4/k4H4 (3.86)
whereH1 = H2+H3 = H4and the rule for combining resistances in parallel has been employed Onceq has been determined, the interface temperatures may be computed:
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Figure 3.9 Series composite wall and its thermal network
T s,1
L1
H1
H3
L 1
H2
L 3
L3
L4
L4
H4
L2
k2
k 3
k4
T2
T2
T1
T1
T s,2
k H1 1
k H2 2
k H4 4
k H3 3 q
Figure 3.10 Series–parallel composite wall and its thermal network
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T s,1
r3
T2 T s,2
h12r L1 2 k L 1 2 k L 2 h22r L3
ln r r( / )2 1 ln r r( / )3 2
L
Hot fluid ,
T h⬁1 1
Figure 3.11 Series composite hollow cylinder and its thermal network
inside and outside experiencing convection is shown in Fig 3.11 The figure includes the thermal network that represents the system The rate ofheat transferq is given by
1/2πh1r1L + ln(r2/r1)/2πk1L + ln(r3/r2)/2πk2L + 1/2πh2r3L (3.89)
Onceq has been determined, the inside surface T s,1, the interface temperatureT2, and the outside surface temperatureT s,2can be found:
T s,1 = T ∞,1 − q 1
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T2 = T ∞,1 − q
1
2πh1r1L +
ln(r2/r1)
2πk1L
(3.91)
T s,2 = T ∞,2 + q 1
made oftwo layers and experiencing convective heating on the inside surface and convective cooling on the outside surface From the thermal network, also shown in Fig 3.12, the rate ofheat transferq can be determined as
1/4πh1r2
1 + (r2 − r1)/4πk1r1r2 + (r3 − r2)/4πk2r3r2 + 1/4πh2r2
3 (3.93)
Onceq has been determined, temperatures T s,1 , T2, andT s,2can be found:
T s,1 T s,1
T s,1
k1
r1
k2
r2
r3
T2
T2 T s,2
T⬁1
q
T⬁2
h14r1 4 k 1 4 k 2 h24r3
1/r1⫺1/r2 1/r2⫺1/r3
q
Cold fluid ,
T h⬁2 2
Hot fluid ,
T h⬁1 1
Figure 3.12 Series composite hollow sphere and its thermal network
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(asperities or peaks), as indicated in Fig 3.13 The gaps ofvoids are usually filled
with air, and the heat transfer at the interface is the sum of solid conduction across the contact points and fluid conduction through the gaps Because ofthe imperfect contact, there is a temperature drop across the gap or interface,∆T c The contact conductanceh c (W/m2· K) is defined as the ratio ofthe heat flux q/A through the
interface to the interface temperature drop:
h c= q/A ∆T
c = R1
c
(3.97) whereq/A is the heat flux through the interface and R
c (m2· K/W), the inverse of h c,
is the contact resistance
The topic ofcontact conductance is ofconsiderable contemporary interest, as reflected by a review paper ofFletcher (1988), a book by Madhusudana (1996),
Figure 3.13 Contact interface for actual surfaces
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and numerous contributions from several research groups Chapter 4 of this book
is devoted exclusively to this subject
3.4.7 Critical Thickness of Insulation
When a plane surface is covered with insulation, the rate of heat transfer always decreases However, the addition ofinsulation to a cylindrical or spherical surface increases the conduction resistance but reduces the convection resistance because
ofthe increased surface area The critical thickness of insulation corresponds to the
condition when the sum ofconduction and convection resistances is a minimum For
a given temperature difference, this results in a maximum heat transfer rate, and the critical radiusr cis given by
r c=
k
2k
wherek is the thermal conductivity ofthe insulation and h is the convective heat
transfer coefficient for the outside surface
The basic analysis for obtaining the critical radius expression has been modified
to allow for:
• The variation of h with outside radius
• The variation of h with outside radius, including the effect of
temperature-dependent fluid properties
• Circumferential variation of h
• Pure radiation cooling
• Combined natural convection and radiation cooling The analysis for a circular pipe has also been extended to include insulation boundaries that form equilateral polygons, rectangles, and concentric circles Such configurations require a two-dimensional conduction analysis and have led to the conclusion that the concept ofcritical perimeter ofinsulation,P c = 2π(k/h), is
more general than that ofcritical radius The comprehensive review article by Aziz (1997) should be consulted for further details
3.4.8 Effect of Uniform Internal Energy Generation
In some engineering systems, it becomes necessary to analyze one-dimensional steady conduction with internal energy generation Common sources ofenergy generation are the passage ofan electric current through a wire or busbar or a rear window defroster in an automobile In the fuel element of a nuclear reactor, the energy is generated due to neutron absorption A vessel containing nuclear waste experiences energy generation as the waste undergoes a slow process ofdisintegration
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T =
2k L2− x2 +
s,2 s,1
s,2 s,1
The maximum temperature occurs atx = k(T s,2 − T s,1 )/2L ˙q and is given by
Tmax= ˙qL2
2k +
k(T s,2 − T s,1 )2
8˙qL2 +T s,2 + T s,1
Ifthe face atx = −L is cooled by convection with a heat transfer coefficient h1and coolant temperatureT ∞,1, and the face atx = +L is cooled by convection with a
heat transfer coefficienth2and coolant temperatureT ∞,2, the overall energy balance
gives
2˙qL = h1(T s,1 − T ∞,1 ) + h2(T s,2 − T ∞,2 ) (3.104)
T s,1
T s,2 q
h T1, ⬁,1 h T2, ⬁,2
x
Figure 3.14 Conduction in a plane wall with uniform internal energy generation
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WhenT s,1 = T s,2, the temperature distribution is symmetrical aboutx = 0 and is
given by
T = ˙q
2k
L2− x2
and the maximum temperature occurs at the midplane (x = 0), which can be
ex-pressed as
Tmax= ˙qL2
WhenT ∞,1 = T ∞,2 = T∞andh1 = h2 = h, eq (3.104) reduces to
ofeq (3.5) for constantk is
1
r
d dr
r dT dr
and the boundary conditions are
T (r = r1) = T s,1 and T (r = r2) = T s,2 (3.109)
T s,1 r1 r2
r
k
T s,2
L
q
Cold fluid ,
T⬁,2 2h
Cold fluid ,
T⬁,1 1h
Figure 3.15 Conduction in a hollow cylinder with uniform internal energy generation