2-95
2-156 A hollow pipe is subjected to specified temperatures at the inner and outer surfaces. There is also heat generation in the
pipe. The variation of temperature in the pipe and the center surface temperature of 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 centerline. 2 Thermal conductivity is constant.
Properties The thermal conductivity is given to be k = 14 W/m°C.
Analysis The rate of heat generation is determined from
3
222
1
2
2
gen W/m750,26
4/)m 17(m) 3.0(m) 4.0(
W000,25
4/)(
=
−
=
−
==
LDD
WW
e
V
Noting that heat transfer is one-dimensional in the radial r direction, the mathematical formulation of this problem can be
expressed as
1gen =+
e
dT
d
2-96
2-157 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.
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.
AnalysisFor one-dimensional heat transfer in the radial r direction with uniform heat generation, the differential equation for
heat conduction in cylindrical coordinate for the wire can be expressed as
1
𝑻wire(𝒓)= 𝑻𝑰+𝒆̇gen
𝟒𝒌wire (𝒓𝟏
The insulation layer does not involve any heat generation, the heat conduction equation in the insulation layer is
𝑑
2-97
C3 and C4 can be expressed as
𝐶3= −𝑒̇gen
2
m3)(0.0025 m)2
2-98
2-99
2-159 A spherical reactor of 5-cm diameter operating at steady condition has its heat generation suddenly set to 9 MW/m3.
The time rate of temperature change in the reactor is to be determined.
Assumptions 1 Heat conduction is one-dimensional. 2 Heat generation is uniform. 3 Thermal properties are constant.
Properties The properties of the reactor are given to be c = 200 J/kg∙°C, k = 40 W/m∙°C, and
= 9000 kg/m3.
Analysis The thermal diffusivity of the reactor is
C W/m40 26
−
k
t
k
r
r
r
2
k
r
r
r
t
2
At the instant when the heat generation of reactor is suddenly set to 90 MW/m3 (t = 0), the temperature variation can be
expressed by the given T(r) = a – br2, hence
+−
=
+
−
=
k
e
brr
r
r
k
e
bra
r
r
r
r
t
T
gen
2
2
gen
22
2
)2(
1
)(
1
2-100
2-160 A cylindrical shell with variable conductivity is subjected to specified temperatures on both sides. The rate of heat
transfer through the shell is to be determined.
Assumptions 1 Heat transfer is given to be steady and one-dimensional. 2 Thermal conductivity varies quadratically. 3 There
is no heat generation.
Properties The thermal conductivity is given to be
)1()( 2
0TkTk
+=
.
Analysis When the variation of thermal conductivity with temperature k(T) is known, the average value of the thermal
conductivity in the temperature range between
21 and TT
is determined from
+
−
+
=
−
=
3
0
12
2
0
12
avg
3
)1(
)(
2
1
2
1
2
1
TTk
TT
dTTk
TT
dTTk
k
T
T
T
T
T
T
r2
T2
r
r1
T1
k(T)
2-101
2-161 A pipe is used for transporting boiling water with a known inner surface temperature in a surrounding 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.
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 0125.0m 2/025.0
1==r
and
m 0155.0m)003.00125.0(
2=+=r
The rate of heat transfer at the pipe’s outer surface can be
expressed as
convcylinder QQ =
21
−TTLrh
TT
2-102
2-162 A metal spherical tank, filled with chemicals undergoing an exothermic reaction, has a known inner surface
temperature. The tank wall has a variable thermal conductivity. Convection heat transfer occurs on the outer tank surface.
The heat flux on the inner surface of the tank 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 The inner and outer radii of the tank are
m 5.2m2/5
1==r
and
m 51.2m)01.05.2(
2=+=r
The rate of heat transfer at the tank’s outer surface
can be expressed as
convsph QQ =
2
21
−TTrh
TT
“GIVEN”
h=80 [W/(m^2*K)] “outer surface h”
r_1=5/2 [m] “inner radius”
r_2=r_1+0.010 [m] “outer radius”
T_1=120+273 [K] “inner surface T”
T_inf=15+273 [K] “ambient T”
k_0=9.1 [W/(m*K)]
beta=0.0018 [K^-1]
“SOLVING FOR OUTER SURFACE TEMPERATURE AND k_avg”
k_avg=k_0*(1+beta*(T_2+T_1)/2)
q_dot_sph=k_avg*r_1/r_2*(T_1-T_2)/(r_2-r_1) “heat flux through the spherical layer”
q_dot_conv=h*(T_inf-T_2) “heat flux by convection”
q_dot_sph+q_dot_conv=0
Thus, the heat flux on the inner surface of the tank is
K387.8)(393
2.51
4
21
2
21
21avg
sph
−
−
−
TT
r
TT
rrk
Q
2-103
Fundamentals of Engineering (FE) Exam Problems
2-163 The heat conduction equation in a medium is given in its simplest form as
0
1
gen =+
e
dr
dT
rk
dr
d
r
Select the wrong
statement below.
(a) the medium is of cylindrical shape.
(b) the thermal conductivity of the medium is constant.
(c) heat transfer through the medium is steady.
(d) there is heat generation within the medium.
(e) heat conduction through the medium is one-dimensional.
2-164 Consider a medium in which the heat conduction equation is given in its simplest form as
t
T
r
T
r
r
r
=
11 2
2
(a) Is heat transfer steady or transient?
(b) Is heat transfer one-, two-, or three-dimensional?
(c) Is there heat generation in the medium?
(d) Is the thermal conductivity of the medium constant or variable?
(e) Is the medium a plane wall, a cylinder, or a sphere?
(f) Is this differential equation for heat conduction linear or nonlinear?
2-165 Consider a large plane wall of thickness L, thermal conductivity k, and surface area A. The left surface of the wall is
exposed to the ambient air at T with a heat transfer coefficient of h while the right surface is insulated. The variation of
temperature in the wall for steady one-dimensional heat conduction with no heat generation is
(a)
−
=T
k
xLh
xT )(
)(
(b)
+
=T
Lxh
k
xT )5.0(
)(
(c)
−= T
k
xh
xT 1)(
(d)
−= TxLxT )()(
(e)
=TxT )(
=TxT )(
2-104
2-105
2-170 The variation of temperature in a plane wall is determined to be T(x)=65x+25 where x is in m and T is in °C. If the
temperature at one surface is 38ºC, the thickness of the wall is
(a) 2 m (b) 0.4 m (c) 0.2 m (d) 0.1 m (e) 0.05 m
38=65*L+25
2-171 The variation of temperature in a plane wall is determined to be T(x)=110-48x where x is in m and T is in °C. If the
thickness of the wall is 0.75 m, the temperature difference between the inner and outer surfaces of the wall is
(a) 110ºC (b) 74ºC (c) 55ºC (d) 36ºC (e) 18ºC
2-172 The temperatures at the inner and outer surfaces of a 15–cm-thick plane wall are measured to be 40ºC and 28ºC,
respectively. The expression for steady, one-dimensional variation of temperature in the wall is
(a)
4028)( += xxT
(b)
2840)( +−= xxT
(c)
2840)( += xxT
(d)
4080)( +−= xxT
(e)
8040)( −= xxT
4080)( +−= xxT
2-106
2-173 The thermal conductivity of a solid depends upon the solid’s temperature as k = aT + b where a and b are constants.
The temperature in a planar layer of this solid as it conducts heat is given by
(a) aT + b = x + C2 (b) aT + b = C1x2 + C2 (c) aT2 + bT = C1x + C2
(d) aT2 + bT = C1x2 + C2 (e) None of them
2-174 Hot water flows through a PVC (k = 0.092 W/mK) pipe whose inner diameter is 2 cm and outer diameter is 2.5 cm.
The temperature of the interior surface of this pipe is 35oC and the temperature of the exterior surface is 20oC. The rate of
heat transfer per unit of pipe length is
(a) 22.8 W/m (b) 38.9 W/m (c) 48.7 W/m (d) 63.6 W/m (e) 72.6 W/m
2-175 Heat is generated in a long 0.3-cm-diameter cylindrical electric heater at a rate of 150 W/cm3. The heat flux at the
surface of the heater in steady operation is
(a) 42.7 W/cm2 (b) 159 W/cm2 (c) 150 W/cm2 (d) 10.6 W/cm2 (e) 11.3 W/cm2
2-107
2-176 Heat is generated uniformly in a 4-cm-diameter, 16-cm-long solid bar (k = 2.4 W/mºC). The temperatures at the center
and at the surface of the bar are measured to be 210ºC and 45ºC, respectively. The rate of heat generation within the bar is
(a) 240 W (b) 796 W b) 1013 W (c) 79,620 W (d) 3.96106 W
2-177 Heat is generated in a 8-cm-diameter spherical radioactive material whose thermal conductivity is 25 W/m.C
uniformly at a rate of 15 W/cm3. If the surface temperature of the material is measured to be 120C, the center temperature of
the material during steady operation is
(a) 160C (b) 280C (c) 212C (d) 360C (e) 600C
2-108