2-21
2-59 The base plate of a household iron is subjected to specified heat flux on the left surface and to specified temperature on
the right surface. The mathematical formulation, the variation of temperature in the plate, and the inner surface temperature
are to be determined for steady one-dimensional heat transfer.
Assumptions 1 Heat conduction is steady and one-dimensional since the surface area of the base plate is large relative to its
thickness, and the thermal conditions on both sides of the plate are uniform. 2 Thermal conductivity is constant. 3 There is
no heat generation in the plate. 4 Heat loss through the upper part of the iron is negligible.
Properties The thermal conductivity is given to be k = 20 W/m°C.
Analysis (a) Noting that the upper part of the iron is well insulated and thus the entire heat generated in the resistance wires is
2-22
2-23
2-61 A large plane wall is subjected to specified temperature on the left surface and convection on the right surface. The
mathematical formulation, the variation of temperature, and the rate of heat transfer are to be determined for steady one–
dimensional heat transfer.
Assumptions 1 Heat conduction is steady and one-dimensional. 2 Thermal conductivity is constant. 3 There is no heat
generation.
Properties The thermal conductivity is given to be k = 2.3 W/m°C.
Analysis (a) Taking the direction normal to the surface of the wall to be the x direction with x = 0 at the left surface, the
mathematical formulation of this problem can be expressed as
0
2
2
=
dx
Td
and
C90)0( 1== TT
])([
)(
−=− TLTh
dx
LdT
k
(b) Integrating the differential equation twice with respect to x yields
1
C
dx
dT =
21
)( CxCxT +=
where
C1
and C2 are arbitrary constants. Applying the boundary conditions give
x = 0:
1221 0)0( TCCCT =→+=
x = L:
hLk
TTh
C
hLk
TCh
CTCLChkC +
−
−=→
+
−
−=→−+=−
)(
)(
])[( 1
1
2
1211
Substituting
21 and CC
into the general solution, the variation of temperature is determined to be
x
x
Tx
hLk
TTh
xT
1.13190
C90
m) 4.0)(C W/m24()C W/m3.2(
C)2590)(C W/m24(
)(
)(
2
2
1
1
−=
+
+
−
−=
+
+
−
−=
(c) The rate of heat conduction through the wall is
W9045=
+
−
=
+
−
=−=−=
m) 4.0)(C W/m24()C W/m3.2(
C)2590)(C W/m24(
)m 30)(C W/m3.2(
)(
2
2
2
1
1wall hLk
TTh
kAkAC
dx
dT
kAQ
Note that under steady conditions the rate of heat conduction through a plain wall is constant.
x
T =25°C
h=24 W/m2.°C
T1=90°C
A=30 m2
L=0.4 m
k
2-24
2-62 A large plane wall is subjected to convection on the inner and outer surfaces. The mathematical formulation, the
variation of temperature, and the temperatures at the inner and outer surfaces to be determined for steady one-dimensional
heat transfer.
Assumptions 1 Heat conduction is steady and one-dimensional. 2 Thermal conductivity is constant. 3 There is no heat
generation.
Properties The thermal conductivity is given to be k = 0.77 W/mK.
Analysis (a) Taking the direction normal to the surface of the wall to be the x direction with x = 0 at the inner surface, the
mathematical formulation of this problem can be expressed as
2
Td
2-25
2-63 In this example, the concepts of Prevention through Design (PtD) are applied in conjunction with the solution of
steady one-dimensional heat conduction problem. The top surface of the plate is cooled by convection, and temperature at the
bottom surface is measured by an IR thermometer. The variation of temperature in the metal plate and the convection heat
transfer coefficient necessary to keep the top surface below 47°C are to be determined.
Assumptions 1 Heat conduction is steady and one-dimensional. 2 Thermal conductivity is constant. 3 There is no heat
generation in the plate. 4 The bottom surface at x = 0 is at constant temperature while the top surface at x = L is subjected to
convection.
Properties The thermal conductivity of the metal plate is given to be k = 13.5 W/m∙K.
Analysis Taking the direction normal to the surface of the wall to be the x direction with x = 0 at the lower surface, the
mathematical formulation can be expressed as
2
Td
2-26
2-64 A series of ASME SA-193 carbon steel bolts of 1 cm thread length are bolted on the upper surface of a metal plate.
The upper surface is exposed to convection with the ambient air. The bottom surface is subjected to a uniform heat flux.
Formulate the temperature profile in the metal plate, and determine the location in the plate where the temperature begins to
exceed 260°C. The compliance of the SA-193 bolts with the ASME Boiler and Pressure Vessel Code (ASME BPVC.IV–
2015, HF-300) is to be determined.
Assumptions1 Heat transfer is steady. 2 One dimensional heat conduction through the metal plate. 3 The bottom surface at x
= 0 is subjected to uniform heat flux while the upper surface at x = L is at uniform temperature. 4 There is no heat generation
in the plate. 5 Thermal properties are constant.
Properties The thermal conductivity of the metal plate is given as 15 W/m·K.
Analysis The uniform heat flux on the bottom plate surface (x = 0) is equal to the heat flux transferred by convection on the
upper surface (x = L):
2-27
2-65 A plane wall is subjected to uniform heat flux on the left surface, while the right surface is subjected to convection and
radiation heat transfer. The variation of temperature in the wall and the left surface temperature are to be determined for
steady one-dimensional heat transfer.
Assumptions 1 Heat conduction is steady and one-dimensional. 2 Temperatures on both sides of the wall are uniform. 3
Thermal conductivity is constant. 4 There is no heat generation in the wall. 5 The surrounding temperature T∞ = Tsurr = 25°C.
Properties Emissivity and thermal conductivity are given to be 0.70 and 25 W/m∙K, respectively.
Analysis(a) Taking the direction normal to the surface of the wall to be the x direction with x = 0 at the left surface, the
mathematical formulation can be expressed as
2=
Td
2-28
2-29
2-67 A 20-mm thick draw batch furnace front is subjected to
uniform heat flux on the inside surface, while the outside
surface is subjected to convection and radiation heat transfer.
The inside surface temperature of the furnace front is to be
determined.
Assumptions 1 Heat conduction is steady. 2 One dimensional
heat conduction across the furnace front thickness. 3 Thermal
properties are constant. 4 Inside and outside surface
temperatures are constant.
Properties Emissivity and thermal conductivity are given to be
0.30 and 25 W/m ∙ K, respectively
Analysis The uniform heat flux subjected on the inside surface
is equal to the sum of heat fluxes transferred by convection and
radiation on the outside surface:
)()( 4
surr
4
0TTTThq LL −+−=
444428
22
K ])27320()[K W/m1067.5)(30.0(
K )]27320()[K W/m10( W/m5000
+−+
+−=
−
L
L
T
T
Copy the following line and paste on a blank EES screen to solve the above equation:
5000=10*(T_L-(20+273))+0.30*5.67e-8*(T_L^4-(20+273)^4)
Solving by EES software, the outside surface temperature of the furnace front is
K 594=
L
T
For steady heat conduction, the Fourier’s law of heat conduction can be expressed as
dx
dT
kq −=
0
Knowing that the heat flux and thermal conductivity are constant, integrating the differential equation once with respect to x
yields
1
0
)( Cx
k
q
xT +−=
Applying the boundary condition gives
:Lx =
1
0
)( CL
k
q
TLT L+−==
→
L
TL
k
q
C+= 0
1
Substituting
1
C
into the general solution, the variation of temperature in the furnace front is determined to be
L
TxL
k
q
xT +−= )()( 0
The inside surface temperature of the furnace front is
K 598=+
=+== K 594m) 020.0(
K W/m25
W/m5000
)0(
2
0
0L
TL
k
q
TT
Discussion By insulating the furnace front, heat loss from the outer surface can be reduced.
2-30
2-68E A large plate is subjected to convection, radiation, and specified temperature on the top surface and no conditions on
the bottom surface. The mathematical formulation, the variation of temperature in the plate, and the bottom surface
temperature are to be determined for steady one-dimensional heat transfer.
Assumptions 1 Heat conduction is steady and one-dimensional since the plate is large relative to its thickness, and the
thermal conditions on both sides of the plate are uniform. 2 Thermal conductivity is constant. 3 There is no heat generation
in the plate.
Properties The thermal conductivity and emissivity are given to be k =7.2
Btu/hft°F and = 0.7.
Analysis (a) Taking the direction normal to the surface of the plate to be
the x direction with x = 0 at the bottom surface, and the mathematical
formulation of this problem can be expressed as
2
Td
x
T
h
Tsky
75°F
2-31
2-69 A series of ASTM B21 naval brass bolts are bolted on the upper surface of a plate. The upper surface is exposed to
convection with air and radiation with the surrounding surface. Formulate the temperature profile in the plate, and determine
if the bolts comply with the ASME Code for Process Piping.
Assumptions1 Heat transfer is steady. 2 One dimensional heat conduction through the plate. 3 The bottom surface at x = 0 is
well-insulated while the upper surface at x = L is subjected to convection and radiation. 4 There is no heat generation in the
plate. 5 Thermal properties are constant.
Properties The emissivity of the plate and bolts is given as 0.3.
Analysis Taking the direction normal to the surface of the plate to be the x direction with x = 0 at the bottom surface, the
differential equation for heat conduction can be expressed as
𝑑2𝑇
2-32
2-70 Chilled water flows in a pipe that is well insulated from outside. The mathematical formulation and the variation of
temperature in the pipe are to be determined for steady one-dimensional heat transfer.
Assumptions 1 Heat conduction is steady and one-dimensional since the pipe is long relative to its thickness, and there is
thermal symmetry about the center line. 2 Thermal conductivity is constant. 3 There is no heat generation in the pipe.
Analysis (a) Noting that heat transfer is one-dimensional in the radial r direction, the mathematical formulation of this
problem can be expressed as
dT
d
2-33
2-34
1
C
2
C
2-72 The convection heat transfer coefficient between the surface of a pipe carrying superheated vapor and the surrounding
air is to be determined.
Assumptions 1 Heat conduction is steady and one-dimensional and there is thermal symmetry about the centerline. 2 Thermal
properties are constant. 3 There is no heat generation in the pipe. 4 Heat transfer by radiation is negligible.
Properties The constant pressure specific heat of vapor is given to be 2190 J/kg ∙ °C and the pipe thermal conductivity is 17
W/m ∙ °C.
Analysis The inner and outer radii of the pipe are
r
dr
Integrating with respect to r again gives
21 ln)( CrCrT +=
where
1
C
and
2
C
are arbitrary constants. Applying the boundary conditions gives
1
loss
1
1
)(
C
Q
rdT =−=
Q
1
2-35
031.0
W4599
1
)()/ln(
2
1
)( 112
loss
2
+−=
rTrr
kL
Q
rT
2-36
2-73 A subsea pipeline is transporting liquid hydrocarbon. The temperature variation in the pipeline wall, the inner surface
temperature of the pipeline, the mathematical expression for the rate of heat loss from the liquid hydrocarbon, and the heat
flux through the outer pipeline surface are to be determined.
Assumptions 1 Heat conduction is steady and one-dimensional and there is thermal symmetry about the centerline. 2 Thermal
properties are constant. 3 There is no heat generation in the pipeline.
Properties The pipeline thermal conductivity is given to be 60 W/m ∙ °C.
Analysis The inner and outer radii of the pipeline are
C
1
2
2
2
r
dr
1
C
and
2
C
can be expressed explicitly as
2,1
TT
−
1
2
11
11221211
2,1 )/ln(
)/()/ln()/(
)( ,
,Trr
hr
k
hrkrrhrk
TT
rT
+
+
++
−
−=
2-37
(b) The inner surface temperature of the pipeline is
C 45.5 =
+
+
−
+
+
++
−
−=
)C W/m150)(m 258.0(
C W/m60
25.0
258.0
ln
)C W/m250)(m 25.0(
C W/m60
)C W/m250)(m 25.0(
C W/m60
C )570(
)/ln(
)/()/ln()/(
)(
22
2
111
11221211
2,1
1,
,Trr
hr
k
hrkrrhrk
TT
rT
(c) The mathematical expression for the rate of heat loss through the pipeline can be determined from Fourier’s law to be
2
loss
)(
rdT
dr
dT
kAQ
−=
2-38
2-39
2-75 Liquid water flows in a tube with the inner surface lined with PVDC lining. The tube outer surface is subjected to a
known uniform heat flux. The tube inner diameter, the tube wall thickness, the water temperature, and the convection heat
transfer coefficient are known. Formulate the temperature profile in the tube wall, and determine if the PVDC lining is in
compliance with the ASME Code for Process Piping.
Assumptions 1 Heat conduction is steady and one-dimensional. 2 Thermal conductivity is constant. 3 There is no heat
generation in the tube wall. 4 The inner surface at r = r1 is subjected to convection while the outer surface at r = r2 is
subjected to uniform heat flux. 5 The PVDC lining is very thin and the temperature gradient in the lining is negligible.
Properties Thermal conductivity of the tube wall is given to be 15 W/m∙K.
AnalysisFor one-dimensional heat transfer in the radial r direction, the differential equation for heat conduction in cylindrical
coordinate can be expressed as
𝑑
2-40