130
HEAT CONDUCTION EQUATION
2–144 In a nuclear reactor, heat is generated in 1-cm-
diameter cylindrical uranium fuel rods at a rate of 4 10
7
W/m
3
.
Determine the temperature difference between the center and
the surface of the fuel rod.
Answer:
9.0°C
(a) Determine the convection coefficient between the wall
and water.
(b) Show that the steady-state temperature distribution has
the form T(x) ax
2
bx c, and determine the values
and units of a, b, and c. The origin of x is shown in the
figure.
(c) Determine the location and value of the maximum tem-
perature in the wall. Could this location be found with-
out knowing a, b, and c, but knowing that T(x) is a
quadratic function? Explain.
2–148 A plane wall of thickness 2L 50 mm and constant
thermal conductivity k 8 W/m·K experiences uniform heat
generation at a rate
e
·
gen
. Under steady conditions, the temper-
ature distribution in the wall is of the form T(x) abx
2
,
where a 80°C and b 2 10
4
°C/m
2
,and x is in meters.
The origin of the x coordinate is at the midplane of the wall.
(a) Determine the surface temperatures and sketch the tem-
perature distribution in the wall.
(b) What is the volumetric rate of heat generation,
e
·
gen
?
(c) Determine the surface heat fluxes
q
·
s
(L) and q
·
s
(L).
(d) What is the relationship between these fluxes, the heat
generation rate and the geometry of the wall?
2–149 Steady one-dimensional heat conduction takes place
in a long slab of width W (in the direction of heat flow, x) and
thickness Z. The slab’s thermal conductivity varies with tem-
perature as k k*/(T* T), where T is the temperature (in K),
and k* (in W/m) and T* (in K) are two constants. The temper-
atures at x 0 and x W are T
0
and T
W
, respectively. Show
that the heat flux in steady operation is given by
q
·
ln
Also, calculate the heat flux for T* 1000 K, T
0
600 K,
T
W
400 K, k* 7 10
4
W/m, and W 20 cm.
2–150 Heat is generated uniformly at a rate of 4.2 10
6
W/m
3
in a spherical ball (k 45 W/m·K) of diameter 24 cm.
The ball is exposed to iced-water at 0°C with a heat transfer co-
efficient of 1200 W/m
2
·K. Determine the temperatures at the
center and the surface of the ball.
2-151 Exhaust gases from a manufacturing plant are being
discharged through a 10-m tall exhaust stack with outer diam-
eter of 1 m, wall thickness of 10 cm, and thermal conductivity
of 40 W/m·K. The exhaust gases are discharged at a rate of
1.2 kg/s, while temperature drop between inlet and exit of the
exhaust stack is 30 °C, and the constant pressure specific heat
of the exhaust gasses is 1600 J/kg·K. On a particular day, the
outer surface of the exhaust stack experiences radiation with
the surrounding at 27 °C, and convection with the ambient air
at 27 °C also, with an average convection heat transfer coeffi-
cient of 8 W/m
2
·K. Solar radiation is incident on the exhaust
stack outer surface at a rate of 150 W/m
2
, and both the emis-
sivity and solar absorptivity of the outer surface are 0.9.
a
T* T
0
T* T
W
b
k*
W
2–145 Consider a 20-cm-thick large concrete plane wall (k
0.77 W/m·K) subjected to convection on both sides with T
1
22°C and h
1
8 W/m
2
·K on the inside, and T
2
8°C and
h
2
12 W/m
2
·K on the outside. Assuming constant thermal
conductivity with no heat generation and negligible radiation,
(a) express the differential equations and the boundary condi-
tions for steady one-dimensional heat conduction through the
wall, (b) obtain a relation for the variation of temperature in
the wall by solving the differential equation, and (c) evaluate
the temperatures at the inner and outer surfaces of the wall.
2–146 Consider a water pipe of length L 17 m, inner
radius r
1
15 cm, outer radius r
2
20 cm, and thermal con-
ductivity k 14 W/m·K. Heat is generated in the pipe material
uniformly by a 25-kW electric resistance heater. The inner and
outer surfaces of the pipe are at T
1
60°C and T
2
80°C, re-
spectively. Obtain a general relation for temperature distribu-
tion inside the pipe under steady conditions and determine the
temperature at the center plane of the pipe.
2–147 A plane wall of thickness L 4 cm has a thermal con-
ductivity of k 20 W/m·K. A chemical reaction takes place in-
side the wall resulting in a uniform heat generation at a rate of
e
·
gen
10
5
W/m
3
. Sandwiched between the wall and an insulat-
ing layer is a film heater of negligible thickness that generates
a heat flux
q
·
s
16 kW/m
2
. The opposite side of the wall is in
contact with water at temperature T
40°C. A thermocouple
mounted on the surface of the wall in contact with the water
reads T
s
90°C.
FIGURE P2–144
.
.