2-61
2-104 A cylindrical fuel rod is cooled by water flowing through its encased concentric tube while generating a uniform
heat. The variation of temperature in the fuel rod and the center and surface temperatures are to be determined for steady one–
dimensional heat transfer.
Assumptions 1 Heat transfer is steady and one-dimensional with thermal symmetry about the center line. 2 Thermal
conductivity is constant. 3 The rod surface at r = ro is subjected convection. 4 Heat generation in the rod is uniform.
Properties The thermal conductivity is given to be 30 W/mK.
Analysis For one-dimensional heat transfer in the radial r direction, the differential equation for heat conduction in
cylindrical coordinate with heat generation can be expressed as
1gen =+
e
dT
d
e
dT
dgen
2-62
2-105 A long electrical resistance wire that is generating heat uniformly is covered with polyethylene insulation.
Formulate the temperature profiles for the wire and the polyethylene insulation. Determine the temperature at the interface of
the wire and the insulation, and the temperature at the center of the wire. Conclude whether the polyethylene insulation for
the wire meets the ASTM D1351 standard.
Assumptions 1 Heat conduction is steady and one-dimensional. 2 Thermal conductivities are constant. 3 Heat generation in
the wire is uniform. 4 There is no contact resistance at the interface of the wire and the insulation, r = r1. 5 At the center of
the wire, r = 0, is a symmetry boundary. 6 The outer surface of the insulation, r = r2, is subjected to convection and radiation.
PropertiesThe thermal conductivities of the wire and the polyethylene insulation are given to be kwire = 15 W/m∙K and kins =
0.4 W/m∙K, respectively.
2-63
Note that ℎcombined =ℎconv +ℎrad. The arbitrary constants C3 and C4 can be expressed as
𝐶3=−𝑒̇gen
2-64
2-106 Heat is generated uniformly in a spherical radioactive material with specified surface temperature. The mathematical
formulation, the variation of temperature in the sphere, and the center temperature are to be determined for steady one–
dimensional heat transfer.
Assumptions 1 Heat transfer is steady since there is no indication of any changes with time. 2 Heat transfer is one–
dimensional since there is thermal symmetry about the mid point. 3 Thermal conductivity is constant. 4 Heat generation is
uniform.
Properties The thermal conductivity is given to be k = 15 W/m°C.
Analysis (a) Noting that heat transfer is steady and one-dimensional in the radial
r direction, the mathematical formulation of this problem can be expressed as
1
gen
2
e
e
dT
d
Ts=80°C
k
egen
2-65
2-66
2-67
2-108 A spherical communication satellite orbiting in space absorbs solar radiation while losing heat to deep space by
thermal radiation. The heat generation rate and the surface temperature of the satellite are to be determined.
Assumptions 1 Heat transfer is steady and one-dimensional. 2 Heat generation is uniform. 3 Thermal properties are constant.
Properties The properties of the satellite are given to be
= 0.75,
= 0.10, and k = 5 W/m ∙ K.
Analysis For steady one-dimensional heat conduction in sphere, the differential equation is
0
1gen
2
2=+
k
e
dr
dT
r
dr
d
r
and
K 273)0( 0== TT
(midpoint temperature of the satellite)
0
)0( =
dr
dT
(thermal symmetry about the midpoint)
2
)K W/m5(6
)K W/m1067.5)(75.0(
428
−
2-68
Copy the following line and paste on a blank EES screen to solve the above equation:
((e_gen*1.25/3+0.10*1000)/(0.75*5.67e-8))^(1/4)=-e_gen*1.25^2/(6*5)+273
2-69
Variable Thermal Conductivity, k(T)
2-109C The thermal conductivity of a medium, in general, varies with temperature.
2-70
2-114 A silicon wafer with variable thermal conductivity is subjected to uniform heat flux at the lower surface. The
maximum allowable heat flux such that the temperature difference across the wafer thickness does not exceed 2 °C is to be
determined.
Assumptions 1 Heat conduction is steady and one-dimensional. 2 There is no heat generation. 3 Thermal conductivity varies
with temperature.
Properties The thermal conductivity is given to be k(T) = (a + bT + cT2) W/m ∙ K.
Analysis For steady heat transfer, the Fourier’s law of heat conduction can be expressed as
dx
dT
cTbTa
dx
dT
Tkq )()( 2
++−=−=
Separating variable and integrating from
0=x
where
1
)0( TT =
to
Lx =
where
2
)( TLT =
, we obtain
T
L ++−= 2
2-71
2-115 A plate with variable conductivity is subjected to specified temperatures on both sides. The rate of heat transfer
through the plate is to be determined.
Assumptions 1 Heat transfer is given to be steady and one-dimensional. 2 Thermal conductivity varies linearly. 3 There is no
heat generation.
Properties The thermal conductivity is given to be
)1()(
0
TkTk
+=
.
Analysis The average thermal conductivity of the medium in this case is simply
the conductivity value at the average temperature since the thermal conductivity
varies linearly with temperature, and is determined to be
K 350)+(500
2
1)(
1–4–
12
0avgave
=
+
+== TT
kTkk
2-116 On one side, a steel plate is subjected to a uniform heat flux and maintained at a constant temperature. On the other
side, the temperature is maintained at a lower temperature. The plate thickness is to be determined.
Assumptions 1 Heat transfer is steady and one-dimensional. 2 There is no heat generation. 3 Thermal conductivity varies
with temperature.
Properties The thermal conductivity is given to be k(T) = k0 (1 + βT).
Analysis For steady heat transfer, the Fourier’s law of heat conduction can be expressed
as
TT
−
T2
k(T)
T1
2-72
2-73
2-118 The thermal conductivity of stainless steel has been characterized experimentally to vary with temperature. The
average thermal conductivity over a given temperature range and the k(T) = k0 (1 + βT) expression are to be determined.
Assumptions 1 Thermal conductivity varies with temperature.
Properties The thermal conductivity is given to be k(T) = 9.14 + 0.021T for 273 < T < 1500 K.
Analysis The average thermal conductivity can be determined using
+
+
)0105.014.9(
)021.014.9(
)( 1200
300
2
1200
300
2
1
TT
dTT
dTTk
T
T
2-74
2-75
2-120 A pipe is used for transporting boiling water with a known inner surface temperature in surroundings of
cooler ambient temperature and known convection heat transfer coefficient. The pipe wall has a variable thermal
conductivity. The outer surface temperature of the pipe is to be determined to ensure that it is below 50°C.
Assumptions 1 Heat transfer is steady and one-dimensional. 2 There is no heat generation. 3 Thermal conductivity varies
with temperature. 4 Inner pipe surface temperature is constant at 100°C.
Properties The thermal conductivity is given to be k(T) = k0 (1 + βT).
Analysis The inner and outer radii of the pipe are
m 015.0m 2/030.0
1==r
m 018.0m)003.0015.0(
2=+=r
The rate of heat transfer at the pipe’s outer surface can be
expressed as
2-76
2-121 A pipe is used for transporting hot fluid with a known inner surface temperature. The pipe wall has a variable
thermal conductivity. The pipe’s outer surface is subjected to radiation and convection heat transfer. The outer surface
temperature of the pipe is to be determined.
Assumptions 1 Heat transfer is steady and one-dimensional. 2
There is no heat generation. 3 Thermal conductivity varies with
temperature.
Properties The thermal conductivity is given to be
k(T) = k0 (1 + βT), α = ε = 0.9 at the outer pipe surface.
Analysis The inner and outer radii of the pipe are
“GIVEN”
h=60 [W/(m^2*K)] “outer surface h”
r_1=0.15/2 [m] “inner radius”
r_2=r_1+0.005 [m] “outer radius”
T_1=423 [K] “inner surface T”
T_inf=273 [K] “ambient T”
T_surr=273 [K] “surrounding surface T”
alpha=0.9 “outer surface absorptivity”
epsilon=0.9 “outer surface emissivity”
q_dot_solar=100 [W/m^2] “incident solar radiation”
k_0=8.5 [W/(m*K)]
beta=0.001 [K^-1]
“SOLVING FOR OUTER SURFACE TEMPERATURE”
k_avg=k_0*(1+beta*(T_2+T_1)/2)
q_dot_cyl=k_avg/r_2*(T_1-T_2)/ln(r_2/r_1) “heat flux through the cylindrical layer”
q_dot_conv=h*(T_2-T_inf) “heat flux by convection”
q_dot_rad=epsilon*sigma#*(T_2^4-T_surr^4) “heat flux by radiation emission”
q_dot_abs=alpha*q_dot_solar “heat flux by radiation absorption”
q_dot_cyl-q_dot_conv-q_dot_rad+q_dot_abs=0
Discussion Increasing h or decreasing kavg would decrease the pipe’s outer surface temperature.
2-77
2-122 A spherical container has its inner surface subjected to a uniform heat flux and its outer surface is at a known
temperature. The container wall has a variable thermal conductivity. The temperature drop across the container wall thickness
is to be determined.
Assumptions 1 Heat transfer is steady and one-dimensional. 2
There is no heat generation. 3 Thermal conductivity varies
with temperature.
Properties The thermal conductivity is given to be
k(T) = k0 (1 + βT).
Analysis For steady heat transfer, the heat conduction
through a spherical layer can be expressed as
21
2
21
21avg
4
TT
r
TT
rrk
Q
−
−
2-78
)1()(
0
TkTk
+=
2-123 A spherical shell with variable conductivity is subjected to specified temperatures on both sides. The variation of
temperature and the rate of heat transfer through the shell are to be determined.
Assumptions 1 Heat transfer is given to be steady and one-dimensional. 2 Thermal conductivity varies linearly. 3 There is no
heat generation.
Properties The thermal conductivity is given to be
)1()(
0
TkTk
+=
.
Analysis (a) The rate of heat transfer through the shell is expressed as
21
TT
−
T1
k(T)
T2
2-79
2-124 A spherical vessel, filled with chemicals undergoing an exothermic reaction, has a known inner surface
temperature. The wall of the vessel has a variable thermal conductivity. Convection heat transfer occurs on the outer surface
of the vessel. The minimum wall thickness of the vessel is to be determined so that the outer surface temperature is 50°C or
lower.
Assumptions 1 Heat transfer is steady and one-dimensional. 2 There is no heat generation. 3 Thermal conductivity varies
with temperature.
Properties The thermal conductivity is given to be k(T) = k0 (1 + βT).
Analysis The inner and outer radii of the vessel are
m 5.2m2/5
1==r
and
)( 12 trr +=
where t = wall thickness
The rate of heat transfer at the vessel’s outer surface can
be expressed as
2-80