3-41
3-60 A wall is made of a composite stainless steel, copper-silicon, and nonmetal plates. A series of ASTM B21 naval
brass bolts are bolted to a nonmetal plate. The upper surface is exposed to convection with air, while the bottom surface is
subjected to a uniform heat flux. Thermal contact resistances exist at the plate interfaces. The total thermal resistance for the
wall for an area of 1 m2 is to be calculated. Determine whether the ASTM B21 bolts are in compliance with the ASME Code
for Process Piping.
Assumptions1 Heat transfer is steady. 2 One dimensional heat conduction through plates. 3 Uniform surface temperatures. 4.
Thermal properties are constant.
Properties The thermal conductivity for the stainless steel plate is k1 = 13 W/m·K, for the copper-silicon plate is k2 = 36
W/m·K, and for the nonmetal plate is k3 = 0.1 W/m·K. The thermal contact conductance between the stainless steel and
copper-silicon plates is hc1 = 20000 W/m2·K, and between the copper-silicon and nonmetal plates is hc2 = 10000 W/m2·K.
Bolt
Air, T∞, h
Stainless steel plate
Copper-silicon plate
Surface 1, T1
Interface 1, hc1
Nonmetal plate
Surface 2, T2
Uniform heat flux
Interface 2, hc2
Analysis The thermal resistances encountered by the heat flow are (Ri1 and Ri2 are contact resistances for interfaces 1 and 2,
respectively)
R1 Ri1
Ti1,a T∞
Ri2
T1 Ti1,b
R2 R4
Ti2,a Ti2,b
R3
T2
3-42
The bolts are in the nonmetal plate, so to determine if they are in compliance with the code, we will calculate the
temperatures Ti2,b and T2. The heat flux through the thermal resistance R4 is
3-43
3-61 A nonmetal plate is bolted on an ASTM A240 904L stainless steel plate by ASTM B211 6061 aluminum alloy
bolts. The upper surface is exposed to convection with air, and the bottom surface is exposed to convection with hot steam.
Thermal contact resistance exists in the interface between the plates. Determine whether the use of the ASTM A240 904L
plate and ASTM B211 6061 bolts complies with the ASME Code for Process Piping.
Assumptions1 Heat transfer is steady. 2 One dimensional heat conduction through plates. 3 Uniform surface temperatures. 4
Thermal properties are constant.
Properties The thermal conductivity for the ASTM A240 904L stainless steel plate is k1 = 13 W/m·K and for the nonmetal
plate is k2 = 3 W/m·K. The thermal contact conductance between the stainless steel and nonmetal plates is hc = 10000
W/m2·K.
3-44
3-45
Generalized Thermal Resistance Networks
3-62C Two approaches used in development of the thermal resistance network in the x-direction for multi-dimensional
problems are (1) to assume any plane wall normal to the x-axis to be isothermal and (2) to assume any plane parallel to the x–
axis to be adiabatic.
3-46
3-65 A wall is to be constructed of 10-cm thick wood studs or with pairs of 5-cm thick wood studs nailed to each other. The
rate of heat transfer through the solid stud and through a stud pair nailed to each other, as well as the effective conductivity of
the nailed stud pair are to be determined.
Assumptions 1 Heat transfer is steady since there is no indication of change with time. 2 Heat transfer can be approximated
as being one-dimensional since it is predominantly in the x direction. 3 Thermal conductivities are constant. 4 The thermal
contact resistance between the two layers is negligible. 4 Heat transfer by radiation is disregarded.
Properties The thermal conductivities are given to be k = 0.11 W/m°C for wood studs and k = 50 W/m°C for manganese
steel nails.
Analysis (a) The heat transfer area of the stud is A = (0.1 m)(2.5 m) = 0.25 m2. The thermal resistance and heat transfer rate
through the solid stud are
W2.2
=
=
=
=
==
C/W 636.3
C8
C/W 636.3
)m 25.0(C) W/m11.0(
m 1.0
2
stud
stud
R
T
Q
kA
L
R
(b) The thermal resistances of stud pair and nails are in parallel
22
2
2
2
(0.004 m )
50 50 0.000628 m
44
0.1 m 3.18 C/W
(50 W /m C)(0.000628 m )
0.1 m 3.65 C/W
(0.11 W /m C)(0.25 0.000628 m )
1 1 1 1 1
3.65 3.18
nails
nails
stud
total
total stud nails
D
A
L
R
kA
L
kA
R
R R R
pp
éù
êú
= = =
êú
êú
ëû
= = = °
×°
×° –
= + = + ¾ ¾® = 1.70 C/W
8C
1.70 C/W
total
T
Q
R
°
D°
= = =
°4.7 W
(c) The effective conductivity of the nailed stud pair can be determined from
C W/m.0.235 =
=
=⎯→⎯
=
)m C)(0.258(
m) 1.0 W)(7.4(
2
TA
LQ
k
L
T
AkQ effeff
T1
Q
Stud
Rstud
T1
T2
L
T2
3-47
3-66E The thermal resistance of an epoxy glass laminate across its thickness is to be reduced by planting cylindrical copper
fillings throughout. The thermal resistance of the epoxy board for heat conduction across its thickness as a result of this
modification is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Heat transfer through the plate is one-dimensional. 3 Thermal
conductivities are constant.
Properties The thermal conductivities are given to be k = 0.10 Btu/hft°F for epoxy glass laminate and k = 223 Btu/hft°F for
copper fillings.
Analysis The thermal resistances of copper fillings and the epoxy board are in parallel. The number of copper fillings in the
board and the area they comprise are
2
2
fillings)copper of(number 333,33
ft) 12/06.0(ft) 12/06.0(
ft 8333.0
ft 8333.0ft) 12/12(ft) 12/10(
==
==
copper
total
n
A
Rcopper
3-48
3-67 A coat is made of 5 layers of 0.15 mm thick synthetic fabric separated by 1.5 mm thick air space. The rate of heat loss
through the jacket is to be determined, and the result is to be compared to the heat loss through a jackets without the air
space. Also, the equivalent thickness of a wool coat is to be determined.
Assumptions 1 Heat transfer is steady since there is no indication of change with time. 2 Heat transfer through the jacket is
one-dimensional. 3 Thermal conductivities are constant. 4 Heat transfer coefficients account for the radiation heat transfer.
Properties The thermal conductivities are given to be k = 0.13 W/m°C for synthetic fabric, k = 0.026 W/m°C for air, and k =
0.035 W/m°C for wool fabric.
Analysis The thermal resistance network and the individual thermal resistances are
C/W 0462.0
m 0015.0
C/W 0009.0
)m 25.1(C) W/m13.0(
m 00015.0
2
8642
2
97531
=
======
=
=======
air
fabric
L
RRRRR
kA
L
RRRRRR
R1 R2 R3 R4 R5 R6 R7 R8 R9 Ro
Ts1
T2
3-49
3-50
3-69 Prob. 3-68 is reconsidered. The effects of the thickness of the wall and the convection heat transfer coefficient on
the outer surface of the rate of heat loss from the kiln are to be investigated.
Analysis The problem is solved using EES, and the solution is given below.
“GIVEN”
width=5 [m]
height=4 [m]
length=40 [m]
L_wall=0.2 [m]
k_concrete=0.9 [W/m–C]
T_in=40 [C]
T_out=-4 [C]
L_sheet=0.003 [m]
L_styrofoam=0.02 [m]
k_styrofoam=0.033 [W/m-C]
h_i=3000 [W/m^2-C]
h_o=25 [W/m^2-C]
“ANALYSIS”
R_conv_i=1/(h_i*A_1)
A_1=(2*height+width-6*L_wall)*length
R_concrete=L_wall/(k_concrete*A_2)
A_2=(2*height+width-3*L_wall)*length
R_conv_o=1/(h_o*A_3)
A_3=(2*height+width)*length
R_total_top_sides=R_conv_i+R_concrete+R_conv_o
Q_dot_top_sides=(T_in-T_out)/R_total_top_sides “Heat loss from top and the two side surfaces”
R_conv_i_end=1/(h_i*A_4)
A_4=(height-2*L_wall)*(width-2*L_wall)
R_styrofoam=L_styrofoam/(k_styrofoam*A_5)
A_5=(height-L_wall)*(width-L_wall)
R_conv_o_end=1/(h_o*A_6)
A_6=height*width
R_total_end=R_conv_i_end+R_styrofoam+R_conv_o_end
Q_dot_end=(T_in-T_out)/R_total_end “Heat loss from one end surface”
Q_dot_total=Q_dot_top_sides+2*Q_dot_end
Lwall
[m]
Qtotal
[W]
0.1
151098
0.12
131499
0.14
116335
0.16
104251
0.18
94395
0.2
86201
0.22
79281
0.24
73359
0.26
68233
0.28
63751
0.3
59800
0.08 0.12 0.16 0.2 0.24 0.28 0.32
60000
80000
100000
120000
140000
160000
Lwall [m]
Qtotal [W]
3-51
ho
[W/m2.C]
Qtotal
[W]
5
54834
10
70939
15
78670
20
83212
25
86201
30
88318
35
89895
40
91116
45
92089
50
92882
60000
65000
70000
75000
80000
85000
90000
95000
Qtotal [W]
3-52
3-70 A typical section of a building wall is considered. The average heat flux through the wall is to be determined.
Assumptions 1 Steady operating conditions exist.
Properties The thermal conductivities are given to be k23b = 50 W/mK, k23a = 0.03 W/mK, k12 = 0.5 W/mK, k34 = 1.0
W/mK.
Analysis We consider 1 m2 of wall area. The thermal resistances are
C/Wm 1.0
C) W/m0.1(
m 1.0
C/Wm 1032.1
0.005)C)(0.6 W/m50(
m 005.0
m) 08.0(
)(
C/Wm 645.2
0.005)C)(0.6 W/m03.0(
m 6.0
m) 08.0(
)(
C/Wm 02.0
C) W/m5.0(
m 01.0
2
34
34
34
25
23b
b
2323
2
23a
a
2323
2
12
12
12
=
==
=
+
=
+
=
=
+
=
+
=
=
==
−
k
t
R
LLk
L
tR
LLk
L
tR
k
t
R
ba
b
ba
a
The total thermal resistance and the rate of heat transfer are
C/Wm 120.01.0
1032.1645.2
1032.1
645.202.0 2
5
5
34
2323
2323
12total
=+
+
+=
+
+
+=
−
−
R
RR
RR
RR
ba
ba
2
W/m125
=
−
=
−
=
C/Wm 0.120
C)2035(
2
total
14
R
TT
q
3-53
3-54
3-55
3-73 A wall is constructed of two layers of sheetrock spaced by 5 cm16 cm wood studs. The space between the studs is
filled with fiberglass insulation. The thermal resistance of the wall and the rate of heat transfer through the wall are to be
determined.
Assumptions 1 Heat transfer is steady since there is no indication of change with time. 2 Heat transfer through the wall is
one-dimensional. 3 Thermal conductivities are constant. 4 Heat transfer coefficients account for the radiation heat transfer.
Properties The thermal conductivities are given to be k = 0.17 W/m°C for sheetrock, k = 0.11 W/m°C for wood studs, and k
= 0.034 W/m°C for fiberglass insulation.
Analysis (a) The representative surface area is
2
m 65.065.01 ==A
. The thermal resistance network and the individual
thermal resistances are
W40.4
C/W 588.6
C)]9(20[
section) m 0.65m 1 a(for 045.0090.0178.6090.0185.0
C/W 178.6
843.7
1
091.29
1111
C/W 045.0
)m 65.0(C) W/m34(
11
C/W 843.7
)m 60.0(C) W/m034.0(
m 16.0
C/W 091.29
)m 05.0(C) W/m11.0(
m 16.0
C/W 090.0
)m 65.0(C) W/m17.0(
m 01.0
C/W 185.0
)m 65.0(C) W/m3.8(
11
21
41
32
2o2
2
3
2
2
2
41
22
=
−−
=
−
=
=++++=
++++=
=⎯→⎯+=+=
=
==
=
===
=
===
=
====
=
==
total
omiditotal
mid
mid
o
o
fiberglass
stud
sheetrock
i
i
R
TT
Q
RRRRRR
R
RRR
Ah
R
kA
L
RR
kA
L
RR
kA
L
RRR
Ah
R
C/W 6.588
(b) Then steady rate of heat transfer through entire wall becomes
m 65.0
m) 5(m) 12(
T1
Ri
R1
R2
R3
R4
R5
T2
3-56
3-57
F/Btuh 62.355
)ft F)(0.1406ftBtu/h 015.0(
ft 12/9
2
4
=
===
kA
L
RR
airholes
3-58
R4
R6
3-75 A composite wall consists of several horizontal and vertical layers. The left and right surfaces of the wall are maintained
at uniform temperatures. The rate of heat transfer through the wall, the interface temperatures, and the temperature drop
across the section F are to be determined.
Assumptions 1 Heat transfer is steady since there is no indication of change with time. 2 Heat transfer through the wall is
one-dimensional. 3 Thermal conductivities are constant. 4 Thermal contact resistances at the interfacesare disregarded.
Properties The thermal conductivities are given to be kA = kF = 2, kB = 8, kC = 20, kD = 15, kE = 35 W/m°C.
Analysis (a) The representative surface area is
2
m 12.0112.0 ==A
. The thermal resistance network and the individual
thermal resistances are
R1
R2
R3
R5
R7
3-59
3-76 In an experiment, the convection heat transfer coefficients of (a) air and (b) water flowing over the metal foil are to be
determined.
Assumptions1 Steady operating conditions exist. 2 Heat transfer is one-dimensional. 3 Thermal properties are constant. 4
Thermal resistance of the thin metal foil is negligible.
Properties Thermal conductivity of the slab is given to be k = 0.023 W/m ∙ K and the emissivity of the metal foil is 0.02.
Analysis The thermal resistances are
L
cond
rad
conv
R
R
R
or
1
elec
rad
1surr
cond
21
conv
11
TT
Aq
R
TT
R
TT
R−
−
−
−
−
=
1surr
21 1
TT
TT
−
−
−
3-60