2-41
𝑇(𝑟1)=− 𝑇∞,1−𝑇∞,2
𝑘
𝑟1ℎ1+ln𝑟2
𝑟1+𝑘
𝑟2ℎ2(𝑘
𝑟1ℎ1+ln𝑟1
𝑟1)+𝑇∞,1
2-42
2-77 A spherical container is subjected to uniform heat flux on the inner surface, while the outer surface maintains a constant
temperature. The variation of temperature in the container wall and the inner 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 surfaces are uniform. 3 Thermal
conductivity is constant. 4 There is no heat generation in the wall. 5 The inner surface at r = r1 is subjected to uniform heat
flux while the outer surface at r = r2 is at constant temperature T2.
Properties Thermal conductivity is given to be k = 1.5 W/m∙K.
2-43
2-78 A spherical shell is subjected to uniform heat flux on the inner surface, while the outer surface is subjected to
convection heat transfer. The variation of temperature in the shell wall and the outer surface temperature 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 in the wall. 4 The inner surface at r = r1 is subjected to uniform heat flux while the outer surface at r = r2 is
subjected to convection.
2-44
2-79 A spherical container is subjected to specified temperature on the inner surface and convection on the outer 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 since there is no change with time and there is thermal
symmetry about the midpoint. 2 Thermal conductivity is constant. 3 There is no heat generation.
Properties The thermal conductivity is given to be k = 30 W/m°C.
Analysis (a) Noting that heat transfer is one-dimensional in the radial r direction, the mathematical formulation of this
problem can be expressed as
0
2=
dr
dT
r
dr
d
and
C0)( 11 == TrT
])([
)(
2
2
−=− TrTh
dr
rdT
k
(b) Integrating the differential equation once with respect to r gives
1
2C
dr
dT
r=
Dividing both sides of the equation above by r to bring it to a readily integrable form and then integrating,
2
1
r
C
dr
dT =
2
1
)( C
r
C
rT +−=
where C1 and C2 are arbitrary constants. Applying the boundary conditions give
r = r1:
12
1
1
1)( TC
r
C
rT =+−=
r = r2:
−+−=−
TC
r
C
h
r
C
k2
2
1
2
2
1
r1
r2
T1
k
T
h
2-45
2-80 A spherical container is used for storing chemicals undergoing exothermic reaction that provides a uniform heat
flux to its inner surface. The outer surface is subjected to convection heat transfer. The variation of temperature in the
container wall and the inner and outer surface temperatures 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 in the wall. 4 The inner surface at r = r1 is subjected to uniform heat flux while the outer surface at r = r2 is
:
1
rr =
2
1
1
1
1)(
r
C
kq
dr
rdT
k−==−
→
k
r
qC
2
1
11
−=
→
)(
2
rdT
−+−=−
C
C
1
1
+
−= T
r
r
h
k
k
r
qC
2
2
2
2
1
12
11
Substituting C1 and C2 into the general solution, the variation of temperature is determined to be
→
−+=+−= T
k
r
r
C
2
1
2
1
1111
−+= T
k
r
2
1
111
2-46
2-81 A spherical container is subjected to uniform heat flux on the outer surface and specified temperature on the inner
surface. The mathematical formulation, the variation of temperature in the pipe, and the outer surface temperature, and the
maximum rate of hot water supply are to be determined for steady one-dimensional heat transfer.
Assumptions 1 Heat conduction is steady and one-dimensional since there is no change with time and there is thermal
symmetry about the mid point. 2 Thermal conductivity is constant. 3 There is no heat generation in the container.
2-47
Heat Generation in a Solid
2-82C Heat generation in a solid is simply conversion of some form of energy into sensible heat energy. Some examples of
2-85C The rate of heat generation inside an iron becomes equal to the rate of heat loss from the iron when steady operating
conditions are reached and the temperature of the iron stabilizes.
2-86C No, it is not possible since the highest temperature in the plate will occur at its center, and heat cannot flow “uphill.”
2-87 Heat is generated uniformly in a large brass plate. One side of the plate is insulated while the other side is subjected to
convection. The location and values of the highest and the lowest temperatures in the plate are to be determined.
Assumptions 1 Heat transfer is steady since there is no indication of any change with time. 2 Heat transfer is one-dimensional
since the plate is large relative to its thickness, and there is thermal symmetry about the center plane 3 Thermal conductivity
is constant. 4 Heat generation is uniform.
Properties The thermal conductivity is given to be k =111 W/m°C.
2-48
2-49
2-50
2-90 Heat is generated in a large plane wall whose one side is insulated while the other side is subjected to convection. The
mathematical formulation, the variation of temperature in the wall, the relation for the surface temperature, and the relation
for the maximum temperature rise in the plate are to be determined for steady one-dimensional heat transfer.
Assumptions 1 Heat transfer is steady since there is no indication of any change with time. 2 Heat transfer is one-dimensional
since the wall is large relative to its thickness. 3 Thermal conductivity is constant. 4 Heat generation is uniform.
Analysis (a) Noting that heat transfer is steady and one-dimensional in x direction, the mathematical formulation of this
problem can be expressed as
gen
2
e
Td
2-51
2-52
2-92 Heat is generated in a large plane wall whose one side is insulated while the other side is maintained at a specified
temperature. The mathematical formulation, the variation of temperature in the wall, and the temperature of the insulated
surface are to be determined for steady one-dimensional heat transfer.
Assumptions 1 Heat transfer is steady since there is no indication of any change with time. 2 Heat transfer is one-dimensional
since the wall is large relative to its thickness, and there is thermal symmetry about the center plane. 3 Thermal conductivity
is constant. 4 Heat generation varies with location in the x direction.
Properties The thermal conductivity is given to be k = 30 W/m°C.
Analysis (a) Noting that heat transfer is steady and one-dimensional in x
direction, the mathematical formulation of this problem can be expressed as
0
)(
gen
2
2
=+ k
xe
dx
Td
where
Lx
eee /5.0
0gen
−
=
and
0
e
= 8106 W/m3
and
0
)0( =
dx
dT
(insulated surface at x = 0)
== 2
)( TLT
30C (specified surface temperature)
(b) Rearranging the differential equation and integrating,
1
/5.0
0
1
/5.0
0
/5.0
0
2
22
/5.0
Ce
k
Le
dx
dT
C
L
e
k
e
dx
dT
e
k
e
dx
Td Lx
Lx
Lx +=→+
−
−=→−= −
−
−
Integrating one more time,
4
)(
/5.0
2
)( 21
/5.0
2
0
21
/5.0
0CxCe
k
Le
xTCxC
L
e
k
Le
xT Lx
Lx
++−=→++
−
=−
−
(1)
Applying the boundary conditions:
B.C. at x = 0:
k
Le
CC
k
Le
Ce
k
Le
dx
dT L0
11
0
1
/05.0
02
2
0
2
)0( −=→+=→+= −
B. C. at x = L:
k
Le
e
k
Le
TCCLCe
k
Le
TLT LL
2
0
5.0
2
0
2221
/5.0
2
0
2
24
4
)(
++=→++−== −−
Substituting the C1 and C2 relations into Eq. (1) and rearranging give
)]/1(2)(4[)( /5.05.0
2
0
2Lxee
k
Le
TxT Lx −+−+= −−
which is the desired solution for the temperature distribution in the wall as a function of x.
(c) The temperature at the insulate surface (x = 0) is determined by substituting the known quantities to be
C314=
−+−
+=
−+−+=
−
−
)]02()1(4[
C) W/m30(
m) 05.0)( W/m10(8
C30
)]/02()(4[)0(
5.0
236
05.0
2
0
2
e
Lee
k
Le
TT
Therefore, there is a temperature difference of almost 300°C between the two sides of the plate.
T2 =30°C
x
k
gen
e
Insulated
L
2-53
2-54
2-95E Heat is generated uniformly in a resistance heater wire. The temperature difference between the center and the surface
of the wire is to be determined.
Assumptions 1 Heat transfer is steady since there is no change with time. 2 Heat
transfer is one-dimensional since there is thermal symmetry about the center line
and no change in the axial direction. 3 Thermal conductivity is constant. 4 Heat
generation in the heater is uniform.
Properties The thermal conductivity is given to be k = 5.8 Btu/hft°F.
Analysis The resistance heater converts electric energy into heat at a rate
4
max k
2-96 A 2-kW resistance heater wire with a specified surface temperature is used to boil water. The center temperature of the
wire is to be determined.
Assumptions 1 Heat transfer is steady since there is no change with time. 2 Heat transfer
is one-dimensional since there is thermal symmetry about the center line and no change in
the axial direction. 3 Thermal conductivity is constant. 4 Heat generation in the heater is
uniform.
Properties The thermal conductivity is given to be k = 20 W/m°C.
Analysis The resistance heater converts electric energy into heat at a rate of 2 kW. The
rate of heat generation per unit volume of the wire is
38
gen
gen
W2000 ====
EE
4
k
r
Ts
ro
110C
r
2-55
2-56
2-99 A cylindrical nuclear fuel rod is cooled by water flowing through its encased concentric tube. The average temperature
of the cooling water is to be determined.
Assumptions 1 Heat conduction is steady and one-dimensional. 2 Thermal properties are constant. 3 Heat generation in the
fuel rod is uniform.
Properties The thermal conductivity is given to be 30 W/m ∙ °C.
Analysis The rate of heat transfer by convection at the fuel rod surface is equal to that of the concentric tube surface:
)()( tube,2,2rod,1,1 ssss TTAhTTAh −=−
2-57
2-100 The heat generation and the maximum temperature rise in a solid stainless steel wire.
Assumptions 1 Heat transfer is steady since there is no change with time. 2 Heat transfer is one-dimensional since there is
thermal symmetry about the centerline and no change in the axial direction. 3 Thermal conductivity is constant. 4 Heat
generation in the heater is uniform.
Properties The thermal conductivity is given to be k = 14 W/mK.
Analysis (a) The heat generation per unit volume of the wire is
RI
E
e
electric,gen
2
2-58
o
2-101 A long homogeneous resistance heater wire with specified surface temperature is used to heat the air. The temperature
of the wire 3.5 mm from the center is to be determined in steady operation.
Assumptions 1 Heat transfer is steady since there is no indication of any change with time. 2 Heat transfer is one-dimensional
since there is thermal symmetry about the center line and no change in the axial direction. 3 Thermal conductivity is constant.
4 Heat generation in the wire is uniform.
Properties The thermal conductivity is given to be k = 8 W/m°C.
Analysis Noting that heat transfer is steady and one-dimensional in the radial r
direction, the mathematical formulation of this problem can be expressed as
1gen =+
e
dT
d
k
dr
2
and
2
2
gen
4
)( Cr
k
e
rT +−=
(b)
Applying the other boundary condition at
o
rr =
,
rr =
2
gen
2
gen
e
e
+=→+−=
180°C
r
ro
2-59
2-60
2-103 A long resistance heater wire is subjected to convection at its outer surface. The surface temperature of the wire is to be
determined using the applicable relations directly and by solving the applicable differential equation.
Assumptions 1 Heat transfer is steady since there is no indication of any change with time. 2 Heat transfer is one-dimensional
since there is thermal symmetry about the center line and no change in the axial direction. 3 Thermal conductivity is constant.
4 Heat generation in the wire is uniform.
Properties The thermal conductivity is given to be k = 15.1 W/m°C.
Analysis (a) The heat generation per unit volume of the wire is
38
gen
gen
W2000 ====
EE