3-21
3-37E A thin copper plate is sandwiched between two layers of epoxy boards. The effective thermal conductivity of the
board along its 9 in long side and the fraction of the heat conducted through copper along that side are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Heat transfer is one-
dimensional since heat transfer from the side surfaces are disregarded 3
Thermal conductivities are constant.
Properties The thermal conductivities are given to be k = 223 Btu/hft°F for
copper and 0.15 Btu/hft°F for epoxy layers.
Analysis We take the length in the direction of heat transfer to be L and the
width of the board to be w. Then heat conduction along this two-layer plate
can be expressed as (we treat the two layers of epoxy as a single layer that is
twice as thick)
QQQ
+=
epoxycopper
93292.0
)(
total
Epoxy
Copper
Ts
½ tepoxy
tcopper
½ tepoxy
Epoxy
3-22
3-38 Two of the walls of a house have no windows while the other two walls have single– or double-pane windows. The
average rate of heat transfer through each wall, and the amount of money this household will save per heating season by
converting the single pane windows to double pane windows are to be determined.
Assumptions 1 Heat transfer through the window is steady since the indoor and outdoor temperatures remain constant at the
specified values. 2 Heat transfer is one-dimensional since any significant temperature gradients will exist in the direction
from the indoors to the outdoors. 3 Thermal conductivities of the glass and air are constant. 4 Heat transfer by radiation is
disregarded.
Properties The thermal conductivities are given to be k = 0.026 W/m°C for air, and 0.78 W/m°C for glass.
Analysis The rate of heat transfer through each wall can be determined by applying thermal resistance network. The
convection resistances at the inner and outer surfaces are common in all cases.
Walls without windows:
C/W 05775.0
C/Wm 31.2
C/W 003571.0
)m 410(C) W/m7(
11
2
2
wall
wall
22
=
=
−
==
=
==
i
i
valueR
L
R
Ah
R
Wall
L
3-23
4th wall with double pane windows:
C/W 023197.0000694.0020717.0001786.0
C/W 020717.0
27303.0
1
5
033382.0
11
5
11
C/W 27303.0267094.0002968.022
C/W 267094.0
m)8.12.1)(C W/m026.0(
m 015.0
C/W 002968.0
m)8.12.1)(C W/m78.0(
m 005.0
C/W 033382.0
m)8.12.1(5)420(
C/Wm 31.2
eqvtotal
eqv
windowwalleqv
airglasswindow
2o2
air
air
22
glass
glass
2
2
wall
wall
=++=++=
=⎯→⎯+=+=
=+=+=
=
==
=
==
=
−
=
−
==
oi RRRR
R
RRR
RRR
kA
L
R
kA
L
R
A
valueR
kA
L
R
Then
W690=
−
=
−
=
C/W023197.0
C)824(
total
21
R
TT
Q
The rate of heat transfer which will be saved if the single pane windows are converted to double pane windows is
W45346905224
pane
double
pane
singlesave =−=−= QQQ
The amount of energy and money saved during a 7-month long heating season by switching from single pane to double pane
windows become
kWh 22,851=h) 2430kW)(7 534.4( == tQQ savesave
Money savings = (Energy saved)(Unit cost of energy) = (22,851 kWh)($0.08/kWh) = $1828
Ri
Rwall
Ro
Rglass Rair Rglass
3-24
3-25
3-40 An engine cover is subjected to convection heat transfer on the inner surface and the outer surface. The thickness of
a thermal barrier coating (TBC) layer applied on the engine cover outer surface is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Heat transfer is one-dimensional. 3 Thermal conductivitiesare constant.4
Thermal contact resistance at interface is negligible.
PropertiesThe thermal conductivities of the stainless steel and the TBC are given to bek1= 14 W/mK andk2 = 1.1 W/mK,
respectively.
AnalysisThe thermal resistances of different layers are
Ah
R
1
conv,1
1
=
(inside surface convection resistance)
L
1
3-26
3-27
3-42 A nonmetal plate and an ASME SB-96 copper-silicon plate are attached together. The bottom surface is subjected
to uniform heat flux. The top surface is exposed to convection heat transfer. Determine the temperatures Ti and T1, and
whether the use of the ASME SB-96 plate complies with the ASME Boiler and Pressure Vessel Code.
Assumptions1 Heat transfer is steady. 2 One dimensional heat conduction through plates. 3 Uniform heat flux on bottom
surface. 4 Uniform surface temperatures. 5 No contact resistance at the interface. 6 Thermal properties are constant.
Properties The thermal conductivity for the ASME SB-96 copper-silicon plate is given as k1 = 36 W/m·K, and for the
nonmetal plate as k2 = 0.05 W/m·K.
Uniform heat flux
Air, T∞, h
ASME SB-96 copper-silicon plate
Nonmetal plate
Surface 1, T1
Interface, Ti
Surface 2, T2
Analysis The thermal resistances encountered by the heat flow are
R1 R2 R3
T2 Ti T1 T∞
𝑅1=𝐿1
3-28
3-29
1− 𝑇𝑖
3-30
Thermal Contact Resistance
3-44C The resistance that an interface offers to heat transfer per unit interface area is called thermal contact resistance,
c
R
.
The inverse of thermal contact resistance is called the thermal contact conductance.
3-45C The thermal contact resistance will be greater for rough surfaces because an interface with rough surfaces will contain
more air gaps whose thermal conductivity is low.
3-46C Thermal contact resistance can be minimized by (1) applying a thermally conducting liquid on the surfaces before they
3-47C An interface acts like a very thin layer of insulation, and thus the thermal contact resistance has significance only for
3-48C An interface acts like a very thin layer of insulation, and thus the thermal contact resistance is significant for highly
3-31
3-32
3-51 Two cylindrical aluminum bars with ground surfaces are pressed against each other in an insulation sleeve. For
specified top and bottom surface temperatures, the rate of heat transfer along the cylinders and the temperature drop at the
interface are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Heat transfer is
one-dimensional in the axial direction since the lateral surfaces of
both cylinders are well-insulated. 3 Thermal conductivities are
constant.
Properties The thermal conductivity of aluminum bars is given to be
k = 176 W/m°C. The contact conductance at the interface of
aluminum-aluminum plates for the case of ground surfaces and of 20
atm 2 MPa pressure is hc= 11,400 W/m2C (Table 3-2).
Analysis (a) The thermal resistance network in this case consists of two
conduction resistance and the contact resistance, and they are
determined to be
C/W 0447.0
/4]m) (0.05C)[ W/m400,11(
11
22
c
contact =
==
c
Ah
R
C/W 4341.0
/4]m) (0.05C)[ W/m(176
m 15.0
2
plate =
==
kA
L
R
Then the rate of heat transfer is determined to be
W142.4=
+
−
=
+
=
=C/W )4341.020447.0(
C)20150(
2barcontacttotal RR
T
R
T
Q
Therefore, the rate of heat transfer through the bars is 142.4 W.
(b) The temperature drop at the interface is determined to be
C6.4=== C/W) W)(0.04474.142(
contactinterface RQT
Ri
Rglass
Ro
T1
T2
Bar Bar
Interface
3-33
3-52 A thin copper plate is sandwiched between two epoxy boards. The error involved in the total thermal resistance of the
plate if the thermal contact conductances are ignored is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Heat transfer is one-dimensional since the plate is large. 3 Thermal
conductivities are constant.
Properties The thermal conductivities are given to be k = 401 W/m°C for copper plates and k = 0.26 W/m°C for epoxy
boards. The contact conductance at the interface of copper-epoxy layers is given to be hc = 6000 W/m2C.
Analysis The thermal resistances of different layers for unit surface
area of 1 m2 are
11
Copper
plate
3-34
3-35
R
3-54 A two-layer wall is made of stainless steel and aluminum plates pressed together. The stainless steel surface is subjected
to uniform heat flux, while the aluminum surface is subjected to convection heat transfer. The surface temperature of the
stainless steel plate is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Heat transfer is one-dimensional. 3 Thermal conductivities are constant.
PropertiesThe thermal conductivities of the stainless steel and aluminum plates are given to bek1= 14 W/mK andk2 = 237
W/mK, respectively. The thermal contact conductance of the stainless steel-aluminum interface with a surface roughness of
about 25μm at an average pressure of 10 MPa is hc = 2900 W/m2K (Table 3-2).
AnalysisThe thermal resistancesof different layers are
Ak
L
R
1
1
1=
(stainless steel plate resistance)
Ah
R
c
1
interface =
L
2
3-36
3-55 The thermal contact conductance for an aluminum plate attached on a copper plate, that is heated electrically, is to be
determined.
Assumptions1 Steady operating conditions exist. 2 Heat transfer is one-dimensional. 3 Thermal properties are constant. 4
Heat transfer by radiation is negligible.
Properties The thermal conductivity of the aluminum plate is given to be 235 W/m ∙ °C.
Analysis The thermal resistances are
kA
L
R=
cond
3-37
3-38
3-57 A thin electronic component is cooled by dissipating heat through a heat sink attached on its top surface. There is
contact resistance at the interface of the electronic component and the heat sink, and the temperature of the electronic
component is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Heat transfer is one-dimensional.3The electronic component maintains a
constant temperature.
PropertiesThe thermal contact conductance at the electronic component/heat sink interface is given as hc = 2000 W/m2K, the
combined convection and radiation thermal resistance of the heat sink is given as 0.3 K/W.
AnalysisThe thermal resistances of different layers are
11
3-39
3-40
3-59 An Inconel® plate covered with a layer of thermal barrier coating (TBC). The plate is exposed to hot combustion gases
with known convection heat transfer coefficient. The temperature of the surface exposed to the hot gases is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Heat transfer is one-dimensional. 3 Thermal conductivities are constant.
PropertiesThe thermal conductivities of the Inconel® and the thermal barrier coating are given to bek1= 25 W/mK andk2 =
1.5 W/mK, respectively. The thermal contact conductance at the interface is given as hc = 3500 W/m2K.
AnalysisThe thermal resistances of different layers are
Ak
L
R
1
1
1=
(Inconel layer resistance)
1